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Two-sided a posteriori estimates for linear elliptic problems
Chapter 6
T w o - s i d e d a posteriori e s t i m a t e s for linear elliptic problems 6.1
B a s i c relations
In this chapter, we consider problems that can be represented in the following general form: to find u E V0 + u0 such that (AAu, Aw) + (l, w) - 0
Vw E V0.
(6.1.1)
Here 1Io is a subspace of a reflexive Banach space V, A is a linear bounded operator acting from V to a HAlbert space U with scalar product (.,.), l E Vo*, and ~4 E ~(U, U) is a self-adjoint operator. We assume that V is compactly embedded in U and the operators A and .4 satisfy the relations
clliyll 2 < (Ay, y) <_ c2]Iy]l2,
Vy E H,
(6.1.2)
and ]]Awl] >_ caliwily,
Vw E Vo,
(6.1.3)
with positive constants cl, c2, and ca. We take the functional I in the form (l, w) - (f, w) + (g, Aw), where f and g are some elements of U. Such a functional is well defined and finite on the elements of V. For our analysis, it is convenient to introduce two more spaces. The quantity (.Ay, y) 1/2 determines a new norm Ill y Ill, which is equivalent to the original norm [lYll - (y,y)l/2. Another equivalent norm is lU Y Ill.= 125
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
126
(j4-1y, y)1/2, where A -1 is the operator inverse to ~4. The spaces Y and Y* contain elements of U equipped with the norms ]l]' ]]1and HI" ]]]., respectively. It is easy to see that (6.1.1) is the Euler relation for the following problem.
Problem 7).
Find u E Vo + uo such that
J(u) =
inf J(u) :--- inf P, vE Vo+uo
where 1
J(v) - -~ ]1]Av ]]]2 +(l, v). On the set (Vo + uo) • Y*, we define the Lagrangian 1
L(v, y*) - ( y * , A v ) - ~ ill Y* III2 +(l, v) and the functional
I*(y*)- inf L(v y * ) vEK
~
I
1 y, 2 ( y * , h u o ) - ~ III III. +(/,uo), l-OO~
y*EQ~, y*r
where
Q~ "- {y* E Y*I(y*,Aw) + (I,w) =O,
VwEVo}.
The problem dual to :P is as follows.
Problem 7)*.
Find p* E Q~ such that I* (p*) -
sup I* (y*) "= sup P*.
The functionals J and ( - I * ) are evidently convex and coercive on V and Y*, respectively. The sets Vo § uo and Q~ are closed affine manifolds. By Theorem 5.4.1, we conclude that there exist u E Vo § uo and p* E Q~ such that J(u) - inf 7~, I* (p*) - sup :P*. (6.1.4) The minimizer u obeys (6.1.1) and the maximizer p* satisfies the relation (Auo - A-~p *, y*) - 0,
Vy* E Q~,
(6.1.5)
6.1. BASIC RELATIONS
127
where Q~) "= {y* E Y*I(y*,Aw)= 0,
Vw E Vo}.
Note that .4An E Q~, so that, setting u0 - u, we obtain 1
I*(Ahu) = (AAu, A u ) - -~ IlI AAu Ill2,+(l, u) _ sup p*.
(6.1.6)
Since ]]I AAu III2-1R h u 1]12,we see that I*(Ahu) = J(u) = inf :P
(6.1.7)
and (cf. (5.4.11)) sup :P* - inf P. The latter relation means that 1 p, 2 1 (p*, A n ) - ~ Ill lU, +(l, u ) = ~ ]lI Au [[I2 -~-(l, U),
which is equivalent to the relation 1 n ( A u , p*) : ~ Ill Au III2
1
III
p,
2 III, -(p*,Au) = o,
(6.1.8)
or simply
p* = MAn
a.ein ~.
(6.1.9)
This is the duality relation for the pair (u, p*). Let v E 110 + u0 and y* E Y* be some approximations of u and p*, respectively. Our goal is to obtain two-sided estimates of the quantities IlI A(v - u) IiI and I]I y* - p* III that are norms of deviations from the exact solutions u and p*. We will call them deviation estimates. Deviation estimates are derived without referring to a particular numerical method used for finding approximate solutions. For this reason, they lead to a posteriori estimates valid for conforming approximations of all types. Below, we show how such estimates are derived by methods of the calculus of variations. First, we establish the following basic result. P r o p o s i t i o n 6.1.1. For any v E Vo + uo and q* E Q~, hold the following
relations: IiI A ( v - u)I~ 2 + Ill q * - P * Ill2, = 2 ( J ( v ) - I'(q')),
(6.1.10)
]lIA(v - u) ]][2 + [][ q, _ p, 1[12,= 2D (Av, q*).
(6.1.11)
128
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
Proof. In view of (6.1.1) and (6.1.5), we have
1
III A(v - u) tO2[ - J(v) - J(u) + ( A A u , A(v - u ) ) + + (l, v - u) - J(v) - J(u),
and 1
iii
q.
-
p.
Ilia, -=
I*(p*) - I*(q*) + (Auo - . A - l p * , p * - q * ) I*(p*) - I*(q*).
Since J(u) = I*(p*), these relations imply (6.1.10). For q* E Q~ the difference J(v) - I*(q*) is equal to D ( A v , q*), so that (6.1.11) follows from (6.1.10). l--I The estimates (6.1.10) and (6.1.11) are valid only for q* E Q~*, which poses some difficulties. Below it is shown how we can overcome this drawback. First, we establish one subsidiary result. Proposition
6.1.2. Let q* E Q~, v E Vo + uo, ~ E R+, and y* E Y*. Then
J(v) - I* (q*) ~_ _< (1 + ~)D(Av, y*) +
(
1+
~ 111
-
tit..
,
)+(iv ,
(6.1.12)
Proof. For any y* E Y*, we have J (v ) - I" ( q" ) - 1(611Av I[Lu +
I]1Y"
]612)+
] q, ili~,-Ill y, +~(Ul
n l2, ) - ( A ~ o q *
-
u0)
Since (l, v - uo) - (q*, A(u0 - v)), we find that J(v) - I* (q* )
1
- ~ (Ui h~ iii: + IlLy* lil~,) - (q*, A~) + 1
+~ (Ill
q,
2
ili, - nl
y,
ni~,) -
1 q, y, 2 = D ( h v , y*) + (y* - q*,Av - A - l y *) + -~ [[[ [1[,"
6.1. BASIC RELATIONS
129
This relation yields (6.1.12) if we note that 2 (y* - q*, Av - A -~ y*) _/3 ]1]hv - A -~ y* [1[2 +/3 -~ Ill Y* - q* [][2,. E] Introduce the quantity
d~(y*)'- ~,~Q; inf Ill q* - y * 1112,9 Then, (6.1.10) and (6.1.12) imply the estimate
1
2 111A(v- u)Ill2<_ (1 + 13)D(Av, y*)+
+
(6.1.13)
2d~(y *)
with v E V0 + u0 and y* E Y*. From (6.1.13), it follows that 1
III A ( v - ,.,)1112< M(v, fi),
V,, e Vo + uo,
,8 e ~ ,
where j~4(v,/3) "- y'ev*inf { ( 1 + fl)D(Av, y * ) + (1+ ~ ) l d ~ ( y * ) } . Let us show that this relation holds as equality. Proposition
6.1.3. For any t3 E I~, 1
III A(v- u)iIi ~- .M(v, Z).
(6.1.14)
Proof. Set y * - ~ p * + ( 1 - ~)AAv. Then D(Av, y*) = 2 A2 INh ( v - u)][]2. Since d~ (y*) _<][] P* - Y* ]U2,= = (1 - A) 2 U[P* - Ahv [[12,- (1 - A) 2 Ill Ah(u - v)[[I,2= = (1 - A) 2 [1[A ( u - v)][i 2, we obtain
~4(~, ~)<
1 ((1
+/x
i1
A/v
/2
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
130
The right-hand side attains its minimal value at A - 1/(1 +/3), which leads to the estimate 1 III i ( v - u)11[2>_ M(v,f~), Vv E V0 + uo, /3 E 11~. Recalling that the inverse inequality has already been established, we arrive at (6.1.14). V1
Proposition 6.1.4.
{1
sup
d~(y*)-
- ~ lUAw
}
lU= -(1,w>- (y*,Aw) .
(6.1.15)
wE Vo
Proof. This assertion is a consequence of the basic duality relation inf P sup P*. Indeed, -
III
-
}
1112, - -
up ,'eQt-v"
-
1 ~?, 2} III III 2 * '
where
Q r - v, := {,* e Y*l,* = ~ , _
y,,
~, e Q;}.
In other words, ~* E Q~ - y * if (7", i w ) = -(l, w) - (y*, Aw),
Vw E Vo.
The right-hand side of this relation is a linear continuous functional. We denote it by 1v and rewrite the relation as follows: (~*, Aw) + (lu, w) = 0 Vw E Vo. Then, Q~ - y* - Q~* and ida(y*)--sup { 1 r]* } . ,, eQr~ - ~ lU 1112, This maximization problem coincides with Problem P* if we set Uo = 0 and l = lv. From this fact, we conclude that inf~evo ~1 Ill Aw I[12+(l v, w) } -
ld~(y*)--
1
= - wevoinf ~ Ill Aw [1[2 +(l, w ) + (y*,Aw) =
sup
{1 -~
Ul Aw III2 - ( z , w) - ( y * ,
.
wE Vo
[:]
131
6.2. TWO-SIDED E S T I M A T E S
Corollary 6.1.1. Propositions 6.1.3 and 6.1.4 lead to the conclusion that the functional A4(v, f~) has a minimax form
M(,,Z) =
( ( 1)( )} inf sup (l+~)D(Av, y*)+ 1 + ~ -(y*,Aw)-g(w) . yEY* wEVo (6.1.16) This relation is crucial for deriving upper and lower bounds of deviations.
6.2
Two-sided
6.2.1
estimates
of deviations
Upper estimates
In view of (6.1.16), we have j~4(v, f~)<_ (1 + ~)D(Av, y*)+ +
(
i)sup{ 1 * 1 + ~ weVo - 2 Hlhw1112 -
},
where y* E Y*. Let A* denote the operator conjugate to A, which is defined by the relation (y*, Aw) = (A*y*, w),
Vw e Yo.
Then
IllllVo*llwilv + Ily*llllAwl[ ~_ (C311111iv(~ Jr-liy*[I)liAwll _~ < ~-~/2 (~xlllllvo. + Ily*ll) INA~ I11.
(6.2.1)
132
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
Hence, the value of the supremum in (6.2.1) is finite and sup
(1 -
IIIAw I[I~ - ( Z , w ) -
(y* Aw)
}
(
I 1 } sup - [1[Aw III2 +IZ+A*y*IIIAwll <
< --
w E Vo
-2
w E Vo
-2
sup ~>0
'
+1/+
2
y'It
-
2
-
-
(6.2.2)
If+A'y*[ 2
Define the functional 1 + h* y ,j2 . 1 + ~1 ) -~[l
u~vt@(v,/3,y*) "- (1 +/3)D(Av, y*)+ By (6.1.16) we see that 1
[][ A ( v - u)]]]2_ .M(v,~) < M~(v,13, y*).
(6.2.3)
Thus, for any y* E Y* and/3 > 0. the functional M~(v,~,y*) is a majorant that provides an upper bound for the deviation v - u evaluated in the natural energy norm. 6.2.2
Lower estimates
Since .M(v, fi) _ sup
inf ((1 4- /3)V(Av, y * ) -
w E V o y . EY*
-
(
1)(1
1+ ~
~ Ill Aw Ill2 +q, w) + (y*, Aw)
)}
,
we have M(~,fl) >
inf
y*EY*
{
(1 + fi)
(1)(~)
~ IIIy* [112,-(Y*, hv)
+(1+/3)~11]Av1112-
-
1+
( 1)(1 1+~
(y*, hw)
}
+
~[l[Awlll 2+(l,w)
)
,
6.2. TWO-SIDED E S T I M A T E S
133
where w is an arbitrary element of Vo. Evidently, this estimate is also valid for f~w, which yields
:nf.{' ~111
"
III.-(y,h(v+~,))
A/I (v, f~) >_ (1 + f~) ,,eY
}+
+(1+,8) (~ UlAv III~ -~ III Aw I!!2 -(l, ~)). Note that inf
y*ffY*
1 y, , } _ Ul Ill~,-(y A(v+w)) > _> y.inf ~y. {1~ III y , 1112.- III y, III. ill A(v + w) III - - 5
1
Ill A(v + w)
Ill~
Thus, we obtain
M(v,~) >
1
(1 +/~) -~. III A(v + w)III ~ + 1
/3
2
+ ~ III A~, III2 - ~. III Aw III - (l, ,.,.,)} = (1 + Z){-(.AAv
A~)
1 + Z III Aw III~ -(l ~)} 2
~
"
Here, w is an arbitrary function in Vo. Therefore, we may replace w by w/(1 + fl). Such a replacement leads to the estimate
89III A(v- ,.v)II1~> Me(,,, ~),
w e Vo,
(6.2.4)
where the minorant is defined by the relation 1
Me(v,,.v) := - ~ III h,.,, III2 - ( A A v , Aw) - (1, w). Remark 6.2.1. Assume that v coincides with u. In this case, 1
1
M e ( U , w) - - ~ Ill Aw lI]~ - ( A A u , A w ) - (l, w) - - ~ II hw ILl2 and, therefore, sup A4e(u, w) = O. wE Vo
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
134
The same is true for the majorant. Indeed, set ~* - .AAu. Then, ~4,(u,~,~)-
(1 + f l ) D ( A u , ~ ) +
( 1)1 1+ ~
~]l+
,
where the first term is zero and
(l, w) + (AAu, Aw)
IZ + A*~*I = ~up so that it is zero as well. Thus,
inf .M~(u, ~, y*) = O.
y*EY*
6.2.3
Estimates of deviations in terms of the dual variable
Let y* E Y* be an approximation of p*. For any q* E Q}', we have
lily -p'Ill2, <(l+'y) llly -q'1112,+ 1 + ~
W -
Ill,,
(6.2.5)
where ~' is a positive number. From (6.2.5), (6.1.10), and (6.1.12), we obtain
IIIy* -p* III,~ < (1 + ~)III y* -q* 1112,+ +2
(11+ ~)
=(1+7)
(
(J(v)-
1 1+-+
1)
,,(q*))= y,_q, I1[ {]1, 2+
+2(1+~) ( 1 + ~1 ) D(Av ,y *) . Therefore, 1-III y*-p* 1112, <- ( 1 + 7 ) 2
1 1 + - ~+
1)1 3~
~ d~(y*)+
+(1+ f ~ )(1 + ~ 1) D(Av'y*)
(6.2.6)
and ~-III y*-p* III,~ < ( 1 + ~ ) 2 +(1 + ~)
1 1 h,y,12 + 1+-+~ ~1 ) ~lz+
(11+ ~)
D(hv, y*).
(6.2.7)
6.3. P R O P E R T I E S OF TWO-SIDED E S T I M A T E S
135
Rewrite this estimate as follows:
89IIIy* - p* 1112,_~ . ~ (y*, ~, ~, ~),
(6.2.8)
where f14~ denotes the right-hand side of (6.2.7). This estimate holds for any y* E Y*, positive parameters/3, 7, and any v E V0 + u0. It is important to outline a certain duality of the estimates (6.2.2) and (6.2.8). In (6.2.2), the majorant contains a "free" function y* that belongs to the space Y* of the dual problem. The minimization of A4r over Y* gives the sharpest upper bound of the deviation evaluated in the energy norm of the primal problem. In (6.2.7), the majorant J~4~ contains a "free" function v, which belongs to the set Vo + u0 of the primal problem. Minimization over this set leads to the sharpest upper bound of the deviation evaluated in the energy norm of the dual problem. Estimates of the deviations in the dual energy norm may be useful in problems arising in continuum mechanics, where the dual variable often has an important physical meaning (e.g., in solid mechanics it is a stress function).
6.3 6.3.1
Properties
of two-sided
estimates
Exactness
Values of ~4o(v, w) depend on the choice of w and values of A/Is(v,/3, y*) depend on f~ and y*. Let us show that w, f~, and y* can always be chosen such that the minorant coincides with the majorant and, consequently, coincides with the energy norm of (v - u). Since [l + A'p*[ - sup (l, w) + (AAu, Aw) _ O, ,~
Vo
IIIhw III
we find that .h4r
- (1 + ~)D(Av, p*) = (1 + f~)
(1 1 ~ lUAv Ill: + ~ ]ll Au Ill2 - ( A A u , Av)
m
1+/~ ][] A(v - u)]]12 2
136
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
and, therefore, 1 ~ Ill A ( v - u)III 2
inf M e ( v , ~ , y * ) y*EY* f~ER+ Similarly,
~) -
Me(~,~-
1
- -~
2 III A(u - v) Ill= - ( .aAv, A(u - v) ) - (l, u - v) 1
-
1
I!1A(u - v) III2 + ( A A ( u - v), A(u - v)) - ~ III A(u - v) Itl2
Thus, the estimate (6.2.4) is exact the in sense that
~sup Vo Me(v,w)-
~1 Ill h ( u -
v)III 2
Let (Yi*}~l and {Voi}~l be two sets of finite-dimensional subspaces. We assume that they are dense in Y* and Vo, respectively. The latter means that, given c > 0 and arbitrary elements y* E Y* and w E V0, one can find a natural number k~ such that inf I1~ - wllv < ~, ~eVoi -
inf III ~* - y* II1< 6, ~*eL* -
Vi > k~. -
(6.3.1)
The methods of creating sequences of finite-dimensional subspaces with above-mentioned property are well known in numerical analysis. In particular, they are represented by regular finite element approximations. Let us show that two-sided bounds can be evaluated precisely by minimizing the m a j o r a n t on {Yi*} and maximizing the minorant on {Voi}. Take a small e > 0,. Then there exists a number k and elements Wk E Vok and p~ E Yo*k satisfying the conditions
Ilwk - ( u - v)llv ___~,
lUp~ - p* Ill, ~ ~.
(6.3.2)
Define two quantities M~-
inf
.Mr
(6.3.3)
f~ER+ and
M~-
sup M e ( ~ , ~ ) . w~ EVok
(6.3.4)
6.3.
PROPERTIES
OF TWO-SIDED
137
ESTIMATES
From (6.3.3) and (6.3.4), it follows t h a t
M ~ _< M ~ ( ~ , ~ , p ~ ) ,
M ~ _> M e ( ~ , ~ ) .
(6.3.5)
1
Let us show t h a t f14~ and f14~ tend to ~[[A(v - u)[[ 2 as k -+ oo. First, we the upper estimate first. We have M~(v,~,pk)
,
-
, + ( 1 + ~1)1L-~lt + A * Pk"12 9 (1 + ~ ) D ( A v , p~)
(6.3.6)
Here
It + A*p~ I - sup (p* - p~' A~o) < ~ eVo IIIA~o III -
(6.3.7)
and I(A v - ..4 -1 Pk,9 A A v D(Av, P k9) - -~
- P k9) --
= 1 (A(v - u ) - A - ~ (p~ - p * ) A A ( v - u ) - (P*k - P * ) ) 2 1 9 p, 2 9 p, -- ~ Ill h ( v - u)Ul 2 + Ill P k Ill, - ( h ( v - u ) , P k -- ). From the latter estimate we see t h a t D(Av,
1 p ~ ) _ ~ Ill A(v - u)
Ill2 + c
1 2 Ill A(v - ~) Ill + ~ 9
(6.3.8)
Combining (6.3.6), (6.3.7), and (6.3.8), we obtain
M~ < Me(~,~,p~)= = (1 +
(1 1~) e) ~ IIIA(v- u)III2 +6 IIIA(v- u)III + ~ + 1 1 + ~6(1 + e) - ~ Ill A(v - u)III 2 -~-C4C-[- O(~),
(6.3.9)
where c4 - ~x (1+ III A(v - u) III)2 . Recall that for any small e there exists a n u m b e r k such t h a t the estimates (6.3.2) hold. Thus, we conclude that
M~
1
~ ~ IIIA ( v -
u)III 2
as
k ~ c~.
(6.3.10)
Remark 6.3.1. Note t h a t the constant ca depends on the norm of ( v - u), so t h a t for a good approximation this convergence is faster t h a n for a bad one.
138
C H A P T E R 6. L I N E A R ELLIPTIC P R O B L E M S Now, let us focus our attention on lower estimates. It is easy to see that 1
M e (~, w~) - - ~ Ul Aw~ Ill2 - (AAv, Awk) - 1 - - ~ Ill Awk Ill + ( A A ( u - v), A w k ) 1 1 = -2 Ill A (u - v) lU~ - ~ III A(wk - (u - v)) III~_>
1
1
_> ~ Ill A ( u - v)Ill2 - ~ c 2 | A ( w k -
( u - v))Ill ~
This implies the estimate 1 Ill A ( ~ -
~)Ill2_> M ~ >_ ~1 IIIA(u- v)lU2 -c5 c2,
(6.3.11)
where c5 is a positive constant dependent on the norm of A. The estimate (6.3.11) shows that
1
M ~ ~ ~ Ill A(u-v)LII
2
as k --+ oo.
(6.3.12)
Remark 6.3.2. Having A/[~ and f14~, one can define the number ~} : - f14~ -> I,
(6.3.13)
which gives an idea of the quality of the error estimation. From (6.3.10) and (6.3.12) it follows that ~k -+ 1,
6.3.2
as k --+ +oo.
Computability
By computability we mean that upper and lower estimates can be computed with any a priori given accuracy. The quantities ~4~ are computed by solving finite-dimensional problems for a quadratic integral functional. Such a task is solvable by well-known numerical methods. However, the quantities A4~ are related to the majorant A4r whose second term is represented by the negative norm. In general, computing such a norm may be very expensive. Below we show that under certain assumptions (which hold in the majority of practically interesting cases), this term is estimated by an explicitly computable quantity. This gives a directly computable majorant that preserves all of the main properties of A4r
139
6.3. PROPERTIES OF TWO-SIDED ESTIMATES Assume that l E U
(6.3.14)
and the variable y* belongs to the set
Q*:={y* eY*l
A'y* EU}
lCU,
and
so that for any w E V0, we have (l + A'y*, w) = (l + A'y*, w). In this case, (l + A'y* w)
II + A*y*I = sup ~,~vo
' III Aw III
< sup - ,o~vo
Ii/+ A*y*II Ilwll III Aw III ii
< II/+ A*y*IIc~ -~
< -
Ii
sup ~
wE Vo iilXWll
< c11c~1111 + A*y*II.
Here cl and c3 are the constants in (6.1.2) and (6.1.3). Now, the majorant is represented in the form
Mr
=
1 ) c2
= (1 + f~)n(Av, y * ) +
1+ ~
71It + A'y* II2
(6.3.15)
where c2 - c-~2c32. If l E U, then p* E Q* and, therefore, inf
y*EQ* ~>0
M~(v,~,y*) < Mr
(1 + r
1
Iii A(u-v)III
2,
where c > 0 may be taken arbitrarily small. Hence, we arrive at the following result. P r o p o s i t i o n 6.3.1. For any v E Vo + uo, ~ E I ~ , and y* E Q*, the following relations holdi
1
III A(u-v)lU=_< M r 1
Ill A ( u - v ) U l
2-
< ir
(6.3.16)
inf Mr
(6.3.17)
y*EQ* f~>0
These relations mean that the majorant M s is also reliable and exact. However, in contrast to j~lr it contains the norm of a Hilbert space U instead of the norm of Vo*. In particular problems, the former is usually
C H A P T E R 6. L I N E A R ELLIPTIC P R O B L E M S
140
given by a directly computable integral, while the latter one is the norm of a Sobolev space with negative index. Obviously, this makes M e much more attractive from the viewpoint of practical applications (see Section 10). To justify this conclusion, we prove that a convergent sequence of upper bounds can be constructed by a sequence of finite-dimensional problems associated with the majorant M e. Assume that {Q~ }~=1 c Q* is a sequence of finite-dimensional spaces, which are dense in Q* in the following sense: for any e > 0 and any q* E Q* one can find a natural number ke such that inf [Iq~n- q*[[Q* ---e, qTeQ~n
Vm > ke,
(6.3.18)
where [[q*[[Q. := [[q*[J + [[i*q* [[. P r o p o s i t i o n 6.3.2. If the spaces ln*xoo t'~ kJk=l are dense in Q 9, then inf M e(v,/3, Ym) - 21 [[[ A ( v - u)[1[2. mlim - ~ u;,eQL
(6.3.10)
r
Proof. Take an arbitrary small r > 0 and find a respective positive integer ke. For m > ke, we have inf M e ( v , ~ , y m ) < M~(v,r y;,~Q;,
)=
BER+
= (1 + e ) D ( A v , p;~) +
1+ ~
7111+ A*p;~[] 2,
(6.3.20)
where p ~ e O~ is an element such that Ilp;~ -P'lie" -- 2e. Since p* e O*, the existence of p ~ follows from (6.3.1S). The first term is easily estimated"
D(Av,p~)-
1 1 ~ I[I Av ][]2 + ~ [][p ~ [][2 - ( p ~ , A v )
__
1 p, , , _ <_ D(Av, p*) + -~ (.A-l(p*~ ),Pine + P*) + (P* -P~nc Av) < 1 1 , p, , p, < ~ [Hh (v - u)) [[[2 + 2 [[I P.~ J[I. (N[ P,~ + [1[. +2 [~ hv [[[), Hence, 1
D(Av,p*~) <_ ~ ]1]A ( v - u))]]12 + # ~ . For the second term we have []/+ A*p~[[ < [[A*(pm~ -P*)[J-< [ [ P ~ - P*J[Q* < #2~,
6.4. ELLIPTIC EQ UATIONS OF THE SECOND ORDER
141
where #1 and #2 are positive constants. Therefore, 1
M e ( v , E , p ~ ) <_ (1 + E)~ Ill A ( v - u)Ill 2 + I
)(e + e') -
IIIA(v u)III2 +o(e'). (6.3.21)
Now, (6.3.20) and (6.3.21)imply (6.3.19).
[:]
+ (~ +
-
In concrete problems, the majorant M e is a quadratic integral functional. Therefore, the numbers M~-
Me(v,/3, y;)
inf
(6.3.22)
/SeR+
can be found by direct minimization procedures. They form a computable sequence of upper bounds such that M~
1 > ~ Ill A ( v - u)Ill 2
as
k --+ c~.
(6.3.23)
Recalling (6.3.12), we conclude that 1
M~ ~ ~ Ill A ( v - u)1l[2<_ M~.
(6.3.24)
The quality of these bounds is estimated by the quantities
~k
"'--
(6.3.25)
z4~'
which are computable and tends to 1 as k -+ +c~.
We can call them
computable effectivity indexes.
6.4
Linear
elliptic
equations
of the second
or-
der Let f~ be a bounded connected domain in IRn with Lipschitz continuous boundary 0f~, V = H1 (f~), U = L2(f~, IEn), and hw be the gradient operator Vw-
(o,l,oxl, ' " , oxn~ . Define a symmetric matrix A aij E L ~
{aij }such that (6.4.1)
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
142 and
cx]~] 2 < aij~i~y _ c2]~i 2,
V~ e lI~~.
(6.4.2)
Then, the spaces Y and Y* are equipped with norms
Ill Y I I I 2 - / A y ' y d x ,
lilY* 1112-/A-ly*'Y *dx.
~2
6.4.1
~2
Dirichlet boundary conditions o
Let V0 - H1(~2). The relation
f AVu. Vw dx + (l,w)
O, Vw E V0,
(6.4.3)
~2
defines a solution of the boundary-value problem div
AVu- f
(6.4.4) (6.4.5)
in ~,
u - uo
on 0~.
The relation (y*, Aw) - ( A ' y * , w) comes in the form
y* . Vwdx = ( - d i v y * , w ) ,
(6.4.6)
fl
where A* - - div and the function div y* is viewed as an element of H -1 (~). The inequality (6.1.3) is the Friederichs inequality O
c~llVwl] >__llw]l,
Vw e Hi(D).
(6.4.7)
Upper estimates of NIv - u III for an approximation v E Vo + uo follow from (6.2.3). We have 1/AV(v-u). 2
V(v-u)dx
< j~4~(v, f~, y*) '
where
Mr
= 1+ Z 2
f n
(Vv -
A - l y * ). ( .AVv - y* ) dx +
1~~ ] div y,_ f ,~
6.4. ELLIPTIC EQUATIONS OF THE SECOND ORDER
143
and
f y* . Vwdx + (f,w} [divy*-f[-
sup (div y* - f, w) = sup fl ~ Vo
Ill A ~ III
~ e Vo
Ill A ~ Ul
Assume that f E L2(f~) and y* E H(f~, div), so that
(f,w)-/fwdx,
/y'.Vwdx--/divy'wdx.
f~
f~
f~
Then, the negative norm has an upper estimate Cn
[divy* - f[ _< ~-1[ ]div
y,
- f]l.
Hence, we arrive at a particular form of the majorant (6.3.15), which yields an estimate
l l AV(v_u) V ( v - u ) d x < 2 tll
f~
l+Of
-
( V v - A-ly*) 9(AVv-y*)dx+
2
f~
1 +Z c~ [[ div y* - f [[~ 2/3 cl
(6.4.8)
valid for any v E Vo + Uo,/3 E I ~ , and y* E H(D, div). Let {Yk*} be finite-dimensional subspaces of Y* such that Yk* E H(12, div) forall k = 1,2,...; dim Yk* -4 +oo as k -+ oo. By (6.4.8) we obtain computable upper bounds
inf {1+/~/
.M~ m
Y~eYk* BER+
2
( V v - A-ly*) . (AVv-y*) dx+
fl
1+~ c~ I[ div y * - f [l~ ~.
2/3 c~
(6.4.9)
J
Lower estimates follow from (6.2.4). We have
12/ AV(v - u) . V(v - u)dx _>Me(v, w), f~
VwE Vo,
(6.4.10)
C H A P T E R 6. L I N E A R ELLIPTIC P R O B L E M S
144 where
1/
M e ( v , w) - --~
A V w . Vw dx -
/
A V v . Vwdx - (f , w).
gt
Let {V Ok} be finite-dimensional subspaces such that
V~k E Vo forall k - 1,2,...; dim Vok -+ +c~
as k -+ c~.
Solving finite-dimensional problems, we find numbers
M~ -
sup M e ( v , wk).
(6.4.11)
w~ E Vok
1 Both sequences M~ and MS tend to ~ Ul~ -
~ Ul~ ~s k -~ ~ , provided that
{Yk*} and {Vok} possess necessary approximation properties (see Section 6.3). Note that if v is a Galerkin approximation computed on Vok, then M e ( v , Wk) -- O. This means that to obtain a sensible lower estimate in this case, one must always use a finite-dimensional subspace that is larger than Vok.
6.4.2
Neumann boundary c o n d i t i o n
If (6.4.5) is replaced by the Neumann boundary condition
v. A V u + F -
0
on
0~t,
(6.4.12)
where v is the vector of unit outward normal to 0~, then the general scheme can be applied if we set
vdx - 01 and define A'y* E Vo* by the relation (A* y*, w) - / y* . V w dx,
Vw e Vo.
Assume that f E L2(D),
F E L2(0gt)
(6.4.13)
6.4. ELLIPTIC EQUATIONS OF THE SECOND ORDER
145
and that the equilibrium condition
I
(6.4.14)
f dx + / F dx - 0
gt
Oft
holds. Then the problem (6.4.4), (6.4.12) has a solution defined by the integral identity
/ A V u . Vw dx + / fw dx + / Fw ds - 0, Ft
~2
Vw E Vo.
(6.4.15)
Oa
In the case under consideration,
(l, w) - / f w dx + / Fw ds a
of~
and
(l +
- /(y*.W
+
~2
O~
In general, ]l + A'y* I is estimated in terms of the norms !] divy* - f IIH-1 and
II y*.u + F IIH-1/2.
However, if we assume that y* possesses a certain regularity, so that y* e Q * ( g t ) ' - {y* e Y* I divy* e L2(~),y*.v e L2(0gt)}, then (l + A'y*, w ) = / ( f - d i v y * ) w d x f~
+ / ( F +y*'u)wds oa
and, therefore, I(l + A'y*, w)l _< _ I] div y* - f [12,a]lwl]2,a + Ilu y* + FI]2,oa]lwll2,oa. (6.4.16) Let the constant ca be defined as
f AVw. Vw dx
i
-
inf
a
2
(6.4.17)
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
146
Since the trace operator is bounded, this constant is finite. (6.4.16) implies the estimate
Therefore,
Ill + A'y*, w)J < <- ca ([[ div y* - /II 22,~ + II~ y*
+ FII2,0n) ~/2
Ul Aw Ill2
and the second term of the majorant is calculated as follows" [l + A*y*J = sup
wEVo
(l + A*y*,w) < I][ h w It[
-
<_ c~ (11 div y* - f I12 2,~
+ I1~'y* +FIJ~,0n) 1/2
The term D(Av, y*) is defined in the same way as in the Dirichlet problem. We see that the distinction in the structure of M e for the two main boundary-value problems manifests itself in different values of cn and in the fact that the Neumann problem majorant contains an extra term IIv y* + FJl2,0n that penalizes violations of the boundary condition (6.4.12).
6.4.3
Mixed
boundary
conditions
Let 012 consist of two measurable nonintersecting parts 0tQ and 02Q, on which different boundary conditions are given:
u-uo
on
0112,
v. A V u + F = 0 on
(6.4.18) 02~.
(6.4.19)
Set V0 : : {v ~ H ~ ( n ) l v - 0
on
0112}
and (h* y *, w) = / y*. Vw dx,
Vw e y0.
Assume that f e L2(12),
F e L2(0212).
Then, the functional I has an integral form
(l,w) = / fwdx + / Fwds. gl 02~2
(6.4.20)
6.4. ELLIPTIC EQUATIONS OF THE SECOND ORDER
147
For any w C Vo, we have (I + A*y*,w>-
f (y* . Vw + fw)dx + / Fwds = o,n
=/(divy*+f)wdx+ /(y*.v+F)wds, ~2
02 ~2
provided that y* possesses an extra regularity, namely, y* e Q*(i2):= {y* e Y" I divy* e L2(12),
y*.v e L2(0212)}.
Note that in view of (6.4.20) the exact solution p* belongs to the set Q*(12), so that the above-made assumption on the admissible set of y* is not restrictive. Now, we obtain
Il _ II divy*
-
.fll~,~ll~ll~,n + lly* 9 v + Fll~,o~ll~ll~,o,n.
Let 7 and 7, be two numbers such that 7>I,
7,>I,
1
1
--F--=I. 9' 7,
Then
I1 ___ (~lldivy*
fll~,~+'Y, Ily* u+Fll~,a~) ~/2
x ~llwll~,a+--Ilwll~,o,a~, Since
I1~11~,~ < c[(~)llVwll ~ --
2,fl~
Vw e go
(6.4.21)
and
2 <_ c2(~,o2~)llwll~,~,n , I1~11~,o~,
w
e Vo ,
(6.4.22)
we find that 1
I7 llwll~,n + --llwll~,a,~ _~ ~',
< ( c~(a)~1 + c~(a, o~a) (~ + c[(a)) -:1 ) ,,Vwi, 2. ,
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
148
Therefore, there exist a positive constant C3 (which depends on ft, 02fl, and 7) such that
f A V w . Vw dx a
1 =
inf
1
2
i
2,ft
,y,
Moreover,
( C ~ ( a ) ~1 + C~(ft, O2ft)(1 + C~(a)) 7--1) c~-~'
C3 _
(6.4.23)
so that the constant C3 has a computable upper bound, provided that the constant C1 (it) in the Friederichs inequality (6.4.21) and the constant C2(fl, 02fl) in the trace inequality (6.4.22) are known. It is worthyof note that these constants depend only on ft and 02ft. Now, we find that I(/+A*Y* , w ) l <_C z ( 7 1 1 d i v y * - fl[ 22,a + 7, IJy*'u + F II~,o,a) 1/2 lU Vw Ill. From this estimate, we obtain I/+A*y
_< 2 f ~ ) <_ < c3 (711divy* - f]l~,a + 7,11Y*" ~' + FII2,o,
<_ c~(a)~ + c~(a,o~a)(1 + c~(a))7 7 1 x
711divy* - fll~,a +
7
7
1
•
Ily*"u + FII2,o~a cF 2.
(6.4.24)
A minimum of the right-hand side of (6.4.24) is attained if 7-~'-1+
Ily*. u + Fll2,o2nCl(a)
il div y* - fil2,ac2(a, o~a)(1 + c[(a))~/~
In this case, II + A*y*l 2
< (C1 (ft)]1 div y* - fl]2,n+ \
+C2(a, 02f~)(1 + C2(fl))l/211y*'u + Fll2,0~a
cf 2. (6.4.25)
6.4. ELLIPTIC EQUATIONS OF THE SECOND ORDER
149
Now, we can present the majorant in two forms. The first one uses (6.4.25), which yields
M e ( v , ~ , y , ) _ 1+2 ~ f (Vv
-
A -1 y *) 9 (AVv
-
y* ) d x +
1 +/~ (C1 (f~)ll div y* - fll2,f~§
+ 2Z
+ c2(a, o~a)(1 + c~(a))~/~lly* ~, + FII2,o~a) 2C l 9
2.
(6.4.26)
If we take 7 - 7* - 2, then the majorant has a simpler form
Me(v ' ~ , y , ) _ l +2 /3 f (Vv
-
A -1 y *) 9 (AVv
-
y* ) d x +
fl
1+ (C~(a) + C~(f~ 02a)(1 + C~(a))) x + 2~ ~ ~ 2
•
2 ) 2,a + IIy, "u + FII2,o2a
9
(6.4.27)
This majorant gives an upper bound of the deviation for any v C V0 + u0, y* C Q*, and ~ > 0. It is easy to see that it vanishes if and only if v = u and y* - AVu. Also, it is obvious that the majorant is continuous with respect to the convergence of v in V and y* in Q"
Remark 6.4.1. The estimate (6.4.26) remains valid for the Neumann problem. 6.4.4
Lower
estimates
Lower estimates for the problems considered follow from (6.2.4). They have a common form:
1 f AV(v - u). V(v - u)dx > lk4e(v, w) 2 '
V w e V0,
f~
where
if
Me(v, w) - --~
f~
AV(w - v) . Vw d x -
f
f~
fw d x -
f
Fw ds.
09f~
Here Vo depends on the type of boundary conditions, and the integral over 02f~ must be eliminated in the case of Dirichldt problem.
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
150
6.5
The linear elasticity problem
Assume that f~ satisfies the same conditions as in the previous section and 0f~ consists of two disjoint parts 01 f~ and 02~ and [01 f~[ > 0. The classical statement of a linear elasticity problem is as follows: to find a tensor-valued function a* (stress) and a vector-valued function u (displacement) that satisfy the system of equations a* = L e
in
f~,
(6.5.1)
div a* = f in
f~,
(6.5.2)
0xfl,
(6.5.3)
u=uo
on
a*u + F = 0 on
02~2.
(6.5.4)
Here f and F are given forces and L - {Lijkm} is the tensor of elasticity constants, which is subject to the conditions Cl1r 2 _< L e ' r
_< C2[r 2,
Vr e M[~s•
(6.5.5)
and
Lijkm = Ljikm = Lk,nij,
Lijk,~ C L~(f~).
(6.5.6)
Henceforth, we assume that f e L2(f~,I~n),
F E L2(02f~,I~n).
(6.5.7)
Then, a generalized solution u E V0 + uo is defined by the identity
/ Le(u) " ~(w) dx + (l,w> - O,
Vw E Vo,
(6.5.8)
f~
where
(t,w) - / . f .wdx + / F.wds. f~
02f~
The existence and uniqueness of u follows from (6.5.5) and the Korn's inequality (see, e.g., [64]). Let v and y* be some approximations of u and a*. Estimates of v - u and y* - a* follow from the general scheme if we set U - L2(D,M~, •
V-Hl(f~,I~n),
Vo - {w E V l w - 0 on 0112},
III y Ill f Ly. f~
Uly* II1.- f f~
y*
151
6.5. THE LINEAR E L A S T I C I T Y P R O B L E M
and identify Av with the symmetric part of the gradient tensor
e(,)- ~1 (w + (w) ~) In this case, (A'y*, w) - f y* " e(w) dx,
(6.5.9)
Vw e Vo,
where two dots stand for the scalar product in ~/~nxn. If y* e Q* "- {y* e Y* I divy* e L2(~2,~•
y*v e L2(02~, I~)},
then (l+ A * y * , w ) - - / d i v y * . w d x + / ( y ' u ) . w d V . ~2
6.5.1
(6.5.10)
82i2
Upper estimates
By (6.2.3) we obtain the upper estimate of the deviation from the exact solution in terms of "strains"" 1 / L~(v - u)" 6(v - u) dx < .Ms (v, ~, y*) 2
-f~
where .Ms (v ~, y*) - 1 +/3 D(ev y*) + 2
l~Z I A'y* + l I:
and 1 /(iL~(v).e(v)L_ly,.y,_ D(e(v), y , ) - -~
2r
y*)dx -
f~
=
f(e(~l-L-~ v*).(Ls(~)-v*)a~. f~
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
152 If y* E Q*, then IA*y * + l l -
sup
(A'y* + l, w>
ill h~ ill
wE Vo
f (y* "~(w) + f . w) dx + f F . w ds :
sup
iii ~ (~) ill
wE Vo
f(f-divy*).wdx+ f - sup
f~
~Vo
(F+y*u).wds
02 ~
III~(w)III
Let Ca be a constant in the inequality
f lwl dx + f Iw[2 ds <_C~lle(w)ll~, ~2
Vw e Vo.
(6.5.11)
02f~
Note that the existence of such a constant follows from the Korn's inequality. By (6.5.11), we obtain
IA*y*+ll< < (lldivy*-fllg+llF+y*vll20,a) 1/2 sup (liwllg+llwl Io~a) 2 1/2 < ,~eVo IIIe(w)III <_ Ca~Y ~/~ (LIdivy *-/ll~ § iIF + Y*"11~2~) ~/~ and arrive at the majorant M e"
M e (e(v), y*) - 1 +2~ f ( c ( u ) - L_ly , )" ( L c ( u ) - y. ) dx+ 1 +r C~ (11 div y* -fll~+llF+y*vllo2n). 2 + 2~cl
(6.5.12)
It has a clear physical meaning. The first term of M e is nonnegative and vanishes if and only if
y* - L~(~). It penalizes violations of the constitutive relation (Hooke's law) (6.5.1). The meaning of the second term is obvious: it contains L2-norms of other two relations (6.5.2) and (6.5.4), which gives errors in the equilibrium equation
6.5. THE LINEAR E L A S T I C I T Y PROBLEM
153
and boundary condition for the stress tensor. Thus, the majorant not only gives an idea of the overall value of the error, but also shows its physically sensible parts. The latter information may be of utmost interest, because it suggests a correct way of finding a better approximation. Let {Yk*} C H 1(~, MI'~x'~ ) be a collection of finite-dimensional subspaces that satisfy the condition (6.3.18). Then, (6.5.12) provides a sequence of computable upper bounds
M ~ ~"
inf { l + / 3 f v" eY~ 2
(L~(v)'c(v)+L-ly*'y
*- 2e(v)'y*)dx
y,
,vl[2 } o n) ,
1 +/3
+ 2/3cl C~
(lldiv
-
Yll
+ IIF + y
(6.5.13)
which tends to the exact value of the error.
6.5.2
Lower
estimates
Lower estimates follow from (6.2.4). We have
1 f L~(v - u)" ~(v - u)dx > .A4e (v, w) 2 '
VwE V0,
where Me(,.,,,-,.,)
-
1 fLe(w)"e(w)dx-fLc(v)"E(w)dx-
2
~2
~2
-fy.wdx-fF.wds. 02 g~
Maximizing the functional Ado on a certain finite-dimensional space Vok C V0, we obtain a computable lower bound A4o-
sup A40(v, wk). wE Vo~
If the spaces Vok satisfy the conditions stated in Section 3, then the sequence of numbers {A4e) tends to (v - u)Ill2 (cf. (6.3.12).
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
154
6.6
Linear elliptic equations der
o f t h e f o u r t h or-
In this section, we consider the problem V. V. (BVVu) - f
Ou
u = 0--~ - 0
in ~2,
(6.6.1)
on O~.
(6.6.2)
Here f~ C I~2, v denotes the outward unit normal to the boundary, and B - "[bijkt} E s 2 1 5 We assume that bijkt -- bjikt = bktij,
o,~ 1,71~ _< B~'r] _< ~lvl 2 , v,7 e M~,,x~,
(6.6.3)
and
.f e L 2 (~),
bijk~ e L ~ (gl).
(6.6.4)
To apply the general scheme, we set U - L 2 (f~, M12x2),
V - H 2 (f~),
Ow
v0-{w e v I w - 0v = ~ on0~}, and define A as the Hessian operator. Now, (6.1.1) has the form
BVVu"
V V w dx = / f w
n
dx
Vw E Vo.
(6.6.5)
n
By B -1 we denote the inverse tensor, which satisfies the double inequality
~-~1,7"1 ~ < B-",7* ",7" _< ~-'1,7"1 ~, V,7* e M~ x~, The spaces Y and Y* are equipped with norms
IIIy II!~- f By" y dx;
Uly* 1112.-f B -~ y*. y* dx,
(l, w) - - / f w dx, f~
and
Qt = {y* e Y* l /y* "VVwdx= / fwdx, vwe Vo}. ~2
~2
(6.6.6)
6.6.
ELLIPTIC EQUATIONS
155
OF THE FOURTH ORDER
Since
IlVVwll > a3JJwJJ2,2,n Vw ~ go, we see that the condition (6.1.3) is satisfied. Problem (6.6.1) and (6.6.2) is associated with two variational problems. Problem P .
Find u E Vo such that J ( u ) = inf J ( v ) , vE Vo
where
1/
J ( v ) - -~
" VVv dx -
BVVv
f
f w dx.
N
Problem 7)* .
Find p* E Q~ such that I*(p*)-
sup I*(q*),
Vq*EQ'~
where I*(q*) - --~1
f
B
--lq* 9 q, dx.
fl
By Proposition 6.1.1, we obtain two basic relations for deviations:
Ill v v ( ~ -
~)ill ~ + IIIq* -P* 1112,- 2(J(v) - I*(q*)),
(6.6.7)
and [] V V ( v - u)II12 + ]1 q* - P "
Ill~, -
2D(VVv, q * ) =
= f (BVVv 9 VVv + B - a q * 9 q* - 2VVv" q*) dx,
(6.6.8)
which hold for any v E V0 and q* E Q~. From (6.1.13) it follows that
1
IIIvv(~- ~)II1~ (1 + f l ) D ( V V v ,
y*)+
+
1+~
1) ~a~(y 1 ,),
(6.6.9)
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
156 where d 2 ( y * ) -
inf
q*eO~
Ill q*-y* III2, Note that
f y* 9VVw dx -
f(div
ft
f~
div y*)w dx,
VwE V0,
so that A* : Y* --+ H-2(f~) is the operator div div. Next, (l + A'y*, w) -
_ / ~ ) dx
vw
(6.6.10)
f~
and, therefore,
f(y* : VVw - fw)dx d~(y*) - I I + A'y* I - sup ~ e Vo
III V V w
III
If y* e H(divdiv, f ~ ) ' - {y* e L2(f~,M~ •
I divdivy* e L2(f~)},
then this quantity is estimated by the relation II + A'y* I < sup - ~
Vo
II div div y* - fll~llwll~ III V V w
III
< -
sup II div div y* - fli. Ilwll. < C~. II div div y* - fll., -~Vo ~llVVwll - ~ <
(6.6.11)
in which Cla is a constant in the inequality Vw E Vo.
Ilwlla __ ClallVVwlla
(6.6.12)
By (6.6.9), we obtain 1
Ill V V ( v - u)[112_< (1 + 13)D(VVv, y*)+ +
+
~l[divdiv
-
fii~, (6.6.13)
In this estimate, y* is an arbitrary tensor-valued function from H(div div, f~) and 13 is a positive real number. However, this is rather demanding in relation to the dual variable y* (which must have square summable div div).
6. 7. RELATIONSHIP WITH OTHER METHODS
157
To avoid technical difficulties that rises from this condition, we estimate the negative norm as follows: [l + A'y*[ < C2---q[Idivy* - rl*[[n + Via II divB* - f[[a. ~1
(6.6.14)
Oll
Here, 77* is an arbitrary vector-valued function from H(div, fl) and Cln is a constant in the inequality [IVw[[a _< Cla[IVVw[[a
Vw
6
V0.
(6.6.15)
Then, we arrive at the estimate 1
[[[ V V ( v - u)[[[2_< (1 + 13)D(VVv, y*)+ -[-
(1 -[-
~
(C2f~[] div
9
- ~ I[a + Cla[I dive* - f]]f~)2
(6.6.16)
in which y* 6 L 2 ( f ~ , ~ • must have square summable divergence and ~* 6 H(div, f~). Note that Cln <_ CnC2n, where Ca is a constant in the Friederichs inequality. In view of this, we obtain a slightly different form of the deviation estimate (cf. [149]): 1
III v v ( ~ -
~)II1~< (1 + 13)D(VVv, y*)+
+ (1 + ~) ~C~a (lldiv y* - o*lla + Calldiv rl* - flla) 2 , (6.6.17) For boundary conditions of other types, the deviation majorants can be derived by arguments similar to those used in Section 4. Lower estimates follow from (6.2.4). We have 1
III VV(v- w)1112>Me(V,W)
w 6 Vo,
(6.6.18)
where 1
Me(v, w).= - ~ III VVw III2 -
/
( B v v v . V V w - fw)dx.
f~
6.7
Relationship with other methods
The majorant ,~4e(V, t3, y*) involves an arbitrary function y*. One way of finding a suitable y* has been discussed. It leads to an auxiliary finitedimensional problem (6.3.3). However, one can also define y* by attracting
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
158
the information encompassed in the approximate solution v. In this section, we are focused on this way. Our aim is to show that such modus operandi leads to a wide spectrum of a posteriori error estimates. For finite element approximations, some of them coincide with a posteriori estimates earlier discussed in Chapter 4. Consider the abstract problem presented in w of this chapter. Assume that (/, w) = (g, w), Vw e V, (6.7.1) where g E U. In this case, p* e Q * :={y* EY* IA*y* e U }
(6.7.2)
and, moreover, p*EQ~:={y*eQ*I(A*y*+g,w)=0,
VweV0}.
(6.7.3)
Let
y~ - AAv.
(6.7.4)
y* - ny~,
(6.7.5)
Setting where II is a certain continuous mapping, we obtain a function y* that can be substituted into the majorant Ad~. 6.7.1
Residual
based
estimates
The simplest way is to set y* - y;,
(6.7.6)
which means that II is the identity mapping of Y* to itself. In this case,
D(Av, y~) = 0 and ,/Via(v, ~, y[) contains only the second term, which upon minimization with respect to/3 gives the first form of the majorant 1
A*
,
which implies the estimate Ig A(v - u) Ill< ]l + A*AAv] - sup (g' w) + (AAv, Aw) ~ Vo iU h ~ iil '
(6.7.8)
If v is obtained by the finite element method (so that v coincides with a Galerkin approximation Uh E Vh := Yoh + Uo), then the right-hand side of (6.7.8) is estimated by the method presented in Chapter 4. As a result, we obtain residual type a posteriori error estimates that involve integral terms associated with finite elements and interelement jumps.
6.7. RELATIONSHIP WITH OTHER METHODS
V
. . . . .
159
L
V " "'-.......
y~
0~
-..
J Figure 6.7.1' Various choices of the dual variable.
6.7.2
E s t i m a t e s b a s e d on the "regularization" dual variable
of t h e
Obviously, the best choice of y* in jM~(v,~,y*) is y* = p* E Q* (cf. 6.7.3). Therefore, if y~ ~ Q* and we know an inexpensive continuous mapping from Y* to Q*, then the image of y~ obtained by such a mapping could be a better approximation of p*. We call such a post-processing procedure the regularization of the dual variable and denote the respective operator by IIreg. By IIreg, we define another element. This situation is presented in Fig. 6.7.1, which is taken from [172]. y~ -- l'IregY~ E Q*
(6.7.9)
and the quantity A4, (v, ~, y{), which gives the second form of the majorant A4~)(v)-
inf
{(1-t-~)D(Av, IIreg(v4Av))-}-
BER+
+
(
1+
~l/+
IIreg(c4Av) }.
(6.7.10)
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
160 Now
]/+A*II~eg(AAv)[- sup (g + A*y~,,w)
weyo
III Aw II]
<- C3111g + A*y~ Jl.
Assume that v - Uh, where Uh is a Galerkin approximation computed for a finite element space Voh C Vo and IIreg - G, where G is a "A-averaging" operator (cf. IV.4). Then 1
D(Av, Hreg(Ahv)) - -~ IIIAAuh -G(.AAUh)lit2,
(6.7.11)
and by (6.7.10) we derive the estimate ]Hih ( u h - u)]i]2__ (1 + f~)I][ A A u h - G(AAuh)tit 2 +
+
( 1+ 1 )~
C~2fIg+
A*G(AAuh)]I
, (6.7.12)
whose right-hand side is easily computable. If we set T - AA, then the first part of (6.7.12) is precisely the error indicator (4.4.3). Of interest is to discuss conditions that make this term dominant. Let the regularity of u and properties of approximation spaces provide that mA(uh - u)]11,~ Ch '~, (6.7.13) where a constant C does not depend on h. Assume that the operator G possesses a superconvergence property (see IV.4), i.e Ill A A u - GAAuh Kit,_ Ch'~+~, ~ > 0. (6.7.14) In this case, the majorant (6.7.10) consists of two terms that have different asymptotic rates. Indeed,
(g + A*GAAuh, w) - (GAAuh - ,AAu, Aw) and we find that
I I + h*GAhuh~ ___titA h u - GAhuh m,_< Ch'~+~. We see that A/[~ ) (Uh) tends to zero as h a, but its second term has a higher convergence rate. Thus, the first term given by (6.7.11) dominates, provided that h is sufficiently small and we conclude that
./~(~) (Uh) '~ D(huh, GAAuh).
(6.7.15)
The quantity D(Auh, GAAuh) is easily computable. Therefore, (6.7.15) gives rise to various practically attractive error indicators.
6. 7. R E L A T I O N S H I P W I T H O T H E R M E T H O D S
6.7.3
Estimates
based
on
the
161
"equilibration"
of the
dual variable Finally, consider the case in which the operator H maps Y* to the set Q}* c Q*. In this case, we define y: = Hy; e Q~.
(6.7.16)
Hence, (A*y~ + l, w) - 0,
Vw E V0,
so that [A*y~ + l] - 0. Now the majorant contains only one term: Ad~ ) (v)
-
(6.7.17)
D(Av, y~).
The meaning of this type error majorants is easy to explain by the paradigm of the linear elasticity problem (see VI.5). In this problem, y* presents a stress field, and the set Q~ consists of the tensor-valued functions satisfying the equilibrium equation (y* " e(w) + f . w ) d x + / g~
F . w ds = O,
Vw E
Vo.
(6.7.18)
02 f~
For this reason, it is natural to call II an equilibration operator and denote it by Heq. The respective majorant has the form
1/(: (v)-
34(~)(v)= ~
Ileq(Le(v)))" (lLe(v)-Ileq(Le(v)))dx. (6.7.19)
f~
Let T*(v) = Le(v) be the stress function associated with :(v). Then, we rewrite ~4~ ) as follows"
lfL-:(r*
:
( v ) - n~qr* (V))'(T* (V)- IIeqT* (v))dx -
~2
1 = 5 Ill T*(V)- H~qT*(V)1112,,
(6.7.20)
The relations (6.7.19) and (6.7.20) show that if a computed stress field T* is post-processed in such a way that it satisfies (6.7.18), then the majorant is given by the error in the Hooke's law, or, what is equivalent, by the U[" I]].norm of the difference between r* and its post-processed image IIeqr*. We
162
C H A P T E R 6. L I N E A R ELLIPTIC P R O B L E M S
recall that this type error estimators for finite element approximations have been discussed in Chapter 4. In general, it is rather difficult to construct an operator IIeq with the More-mentioned projection properties. Instead, one may try to use an operator IIeq, which provides an approximate equilibration. In thise case, the second term of the majorant .h4~(V,~,IIeqT*) does not vanish and should be taken into account. 6.7.4
A priori
projection
type
error
estimates
Now, our goal is to show that classical a priori projection type error estimates also follow from the general deviation estimate (6.2.3). Let Uh E Vh be a Galerkin approximation of u. We have
]]]h(u--uh) ]0]___2(1--F.~)D(huh, y*)~- (1-t-~) ]A'y* +/0(~6.7.21) Set here y* - .Ahvh, where Vh is an arbitrary element of Vh. Then, i A'y* + l[ -
sup (y* - p*' hw) = we Vo ]][ hw ]]] = sup e v0
(AA(vh - u), Aw) JiBh w lil
< III A (~ h - ~) III.
It is easy to see that (6.7.22)
D(Auh,.AAvh) - J(vh) - J(uh). Indeed, 1
1
D(Auh, AAvh) = -~(AAvh, AVh) + (l, vh) - -~(AAuh, AUh) - (l, uh)+ + (AAuh, h ( u h - Vh))+ (l, uh --Vh). Since Uh E Vh is a Galerkin approximation, the last two terms vanish and we arrive at (6.7.22). Recalling (6.1.10), we find that 6]]A ( u h - u)I]]2= 2(J(uh) - J(u)),
161A(vh -u)I]62- 2(J(vh) - J(u)). Therefore, 2D(Auh,
AAvh)
-
2(J(vh)
- J(u))
- 2(J(uh)
- J(u))
-
=Ui h ( ~ h - ~)iil ~ -ill h ( ~ h - ~)ill ~
6.8.
163
INDETERMINACYINTHEDATA
Now, (6.7.21) leads to the estimate
(2 + f~)III A(u- Uh)III~_ (1+,6) Ill A(vh- u)II~+ (1+ 1~ p), Ill A(vh-~)Ill ~, \ which shows that
III A ( u -
Uh)II1~_<
(1
1 )
+ /~(2 + Z)
Ill h(~ - ,h)Ill ~ 9
Since/3 is an arbitrary positive number, we arrive at the projection type error estimate (cf. 2.3.14) [1[A ( u
-
Uh) ][I< --
inf
vhEVh
Ill A(u - Vh) Ul.
(6.7.23)
Finally, we note that the general deviation estimate (6.2.3) also implies another projection type error estimate. Let us set v = Uh, y * -- Yh* o= ,AVUh and use (6.1.13). Since D ( A u h , y~) - O,
we have
iii h (~ h - ~) Ul: _< lil yh* - q* iU,~ Vq* e Q r. From here, it follows the estimate ]]] A(u - Uh)]]] < --
inf
q*EQ~
]]] y~ -- q* ]lJ,,
(6.7.24)
which is in a sense dual to (6.7.23). It shows that an upper bound of the error is given by the distance (in the space Y*) between y~ (which is a dual counterpart of the Galerkin approximation Uh) and the set Q~ containing the exact solution p* of Problem 7~*.
6.8
Error
estimates
determinacy 6.8.1
General
taking in the
account
of the
in-
data
concept
In the majority of applied problems, exact values of the data are unknown. For example, in linear elasticity problems material constants may be known
164
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
only approximately, so that instead of exact numbers we have some intervals. This indeterminacy must be taken into account in the error analysis especially if the influence of the arising additional errors is significant. Below we consider this question by the paradigm of the problem A*AAu = l in ~,
u = uo on Off.
(6.8.1)
where the operator Jt and the functional l are defined with some indeterminacy. Their actual values are unknown. Instead, we know the sets of admissible data. Formally, this means that A E/~A C L:(U, U),
(6.8.2)
l E/it C Vo*,
(6.8.3)
where/~A and/~z are bounded and connected sets that take into account possible variations of the data. It is easily observed that the majorant M s and the minorant M e derived in Section 6.2 explicitly depend on A and I. Because of this fact, functional type majorants and minorants can serve as numerical tools naturally adopted to error estimation if data of a boundary-value problem are not exactly determined. Below, we show how such an estimate can be obtained. To make the exposition clearer, we demonstrate this with the paradigm of the problem (6.4.4)-(6.4.5). Possible generalizations of these results to other linear elliptic equations are rather obvious. In essence, they require choosing proper spaces V and Y and finding the constant C(fl, A), which also depends on properties of the operator A. First note that the solutions of problems with above data form the set T(U~t, Ut) " - { ~ E
Vo + u o I
satisfies (6.8.1) for some~ ,4 EhCA and l E/4t S
Let v E Vo+uo be an approximation of an unknown exact solution. Since the data are indeterminate, the error estimation problem comes in two different forms. The first problem is to find the quantity 2i n em
(v,T)-
inf ~1 [[A(v -
SET
u)l[ 2
(6.8.4)
where the multiplier 1/2 is present for convenience. The quantity e m i n measures the distance between v and the set T. It is equal to zero if v satisfies (6.8.1) for some pair (A,l) E /~A x / ~ . This quantity provides a lower bound of the true error or the error in the best-case situation when the exact solution is the function of T closest to v.
6.8. I N D E T E R M I N A C Y I N T H E D A T A
165
Another problem is to find the quantity 2 (v T ) - s u p ~1 IIA(v-~)] [2 , emax ~ET ,
(6.8.5)
which shows an upper bound of the error. It takes into account computational errors and errors caused by indeterminacy and shows the error in the worst-case situation when the exact solution is an element of T that is most distant of v. This quantity is always positive and its value gives an idea of the accuracy limit dictated by the effect of indeterminacy in the data. Thus,
emin(V,T) <~_e(v) ~__emax(v,T))
(6.8.6)
where the actual error e(v) is principally unknown and we may only hope to find its bounds. In general, the exact values of emin and emax could hardly be found. However, using functional type a posteriori estimates, one can find their computable bounds. Indeed, the majorant jt4~ and the minorant M e explicitly depend on ,4 and l, which opens a way for computing errors caused by the indeterminacy in values of the problem data. Below we show how such an account can be performed. Assume that the set T is known. Our aim is to find practically computable numbers ee(v , T) and e~(v, T) such that for any v E V0 + u0 the following relations hold:
ee(v , T) ~ emin(V,T) < emax(V,T) < e~)(V,T).
(6.8.7)
These numbers give bounds of the total error that is caused by the computational inaccuracy and data indeterminacy. It is worth noting that such type estimates can also be applied to differential equations with random coefficients.
6.8.2
U p p e r b o u n d of t h e
error
In the subsequent text, we take the problem (6.4.4)-(6.4.5) as a basic example. In this case, V = H I (12), V0* = H-l(i2),
Y
=
L2 (fl, I~n),
Av=Vv,
. - ~ ~(fl), Vo.-
A*y*=-divy*,
and ,4 is a mapping given by the relation y*(x) --+ A(x)y*(x), where A(x) is a symmetric positive definite matrix.
166
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
Assume that the indeterminacy in the coefficients of the differential equation (6.8.1) is described by fixing some "mean" elements Ao E L~(fl;1M~ •
and
l0 E L2(~)
and defining bounds of possible variations. In this case, the elements of the sets HA and/~t are represented by elements of the following two functional sets:
HA := {A E L ~ ( g ~ ; M ~ • H I "-- { f E L ~ ( ~ ) i f -
Ao + r
E E E},
fo + (f~o, ~o E .7:},
where g' := {E E L ~176 (~; h~s xn) I [I IEI II~,n .T" := {~o E L2(~)
_ 1},
111~112,~_< 1}.
Here, ~ and 5 are small parameters characterizing the range of indeterminacy. Henceforth, we assume that the parameter ~ is small enough, so that the problems remain uniformly elliptic for all possible data. Formally, this means that the condition cxI~l 2 < Ao~-~ < v2I~I2,
v~ e ~",
(6.8.8)
implies similar estimates for all A C/4A. Since IE~. ~l _ IEI IS[2, we find that A~. ~ > Ao~. ~ - ~IEI I~12 _ (cx - ~)1~12, A~'. ~" < Ao~. ~"+ ~IEI I~12 _< (c2 + ~)1~12.
(6.8.9) (6.8.10)
From here, it is clear that e must be subject to the condition < cl.
(6.8.11)
For the inverse matrix, we have c~lI~[ 2 _ Aol~ 9~ _ c~-1]~[2, (c2 + ~)-1 ]~[2 ___A - I ~ . ~ _ (cl - ~)-1 i~[2, where A E/4,4. To use (6.4.8), we must estimate the functional 1 (A -~ A'y* , A'y* ) - ( A v ,y* I(AAv, A v ) + -~ ) D(Av, y*) "- -~
(6.8.12) (6.8.13)
6.8. I N D E T E R M I N A C Y I N
167
THEDATA
for any A E Ha. Denote the unit matrix by IT. Then A -1 - (Ao(lI + r
-1 - (]I + r
-1 A o 1.
Note that
~IAolE{ < ~llAoXl IEI < ~c-~~ < 1, and, therefore, (]I + r
where B - A o l E . Hence,
-1 = I[ + ~ (-1)Jr j=l
A - l y *. y* - (I + r
* . y* = oo
= A o l y * 9 y* _ e B A o l y * . y* + E ( - 1 ) J e J B J A o l y
*. y*.
j'-2
Since E E t;, we have BJAoly
. y* dx <_
n
/IAo~{ j+l IEI~ ly*l 2 dx
<_ c~-(w+~)lly*ll 2
f~
and A - l y , . y, dx < _ / ( A o l y *. y* - e B A o l y * . y*) dx+
ft _b (~(_l)J~JCl(J+l)) " j = 2 ,y,2 Moreover, A V v . V v dx - / ( A o V v .
ft
V v + e E V v . Vv)dx.
ft
From these relations, we find that the first term of the duality error majorant explicitly depends on ~. It is estimated as follows" D ( V v , y*) <_ D0(Vv, y*) + c / ( E V v .
V v - B A o l y *. y*) dx+
fl
1
y , i 2,
e j=2
j
(6.8.14)
168
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
where D o ( V v , y*) -
/(1 -~AoVv.
1
V v + -~Aoly* . y
,-
Vv.
y,)
dx.
f~
Since all the matrices are symmetric, we have E V v . V v - A o l E A o l y * . y* - E ( V v - A o i y * ) 9 (Vv + A o l y * ) .
Now, (6.8.14) implies the estimate D ( V v , y*) <_ Do(Vv, y*) + ~
~f
E ( V v - A o l y ). (Vv + A o l y ) d x +
*
e cl
9
1 ,i 2 2(6+cl)11Y I 9 (6.8.15)
Recall that (see (6.3.17)
1 III V ( v - u ) I I I 2 -
inf
y*EQ* f~>O
.M$(v,~,y*).
(6.8.16)
For the problem considered, we have UlV(v - u)III 2 -
f(Ao
+ ~E)V(v - u). V(v - u ) d x >_
f~
>_ ( ~
-
~)IIV(v
-
u)ll ~ .
(6.8.17)
Therefore, 2 emax
( v , T ) - s u p l ~ IIV(V -- ~)II 2 < 1 sup III V(v - ~) II1~= JeT -- 2(Cl -- e) ~eT 1 = sup inf M e (v, f~, y*). (c~ - ~) a c t y" eQ" ~>o
By definition, T (b/A,//f) is a subset of V0 + uo that contains solutions of the boundary-value problems (6.8.1) with A-
Ao + ~E
and
l-.fo+&p.
6.8. INDETERMINACY IN THE DATA
169
Since any such problem has a unique solution, we can replace sup by sup. (A,f)
Hence, we obtain CI
-
-
g) emax(V 2 ,T) _ sup
inf ~4r y*) _< ~eT Y*~Q* 3>o < inf sup ~ / r (v, 3, y*)--- y*EQ* fiET f~>o
=
A ) 2(~' /3 l+f~)D(Vv, y*)+ (1 +f~)C2
inf. sup
]Idiv Y* - f ] l 2}"
y* EQ AEI,4A
From here, we deduce the main estimate
e2m~(v, T) __ 1
(1 + 3) sup D(Vv, y*)+
1+
C1
AEblA
Cl -- C
%.
+ (1 +/3) sup C 2(~2, A) sup i] div y* - .fi]2
23
.fEll/
AElgA
(6.8.18) J '
which is valid for any y* E Q* and f~ > 0. The first term on the right-hand side in (6.8.18) is estimated by (6.8.15). Let us consider the second one. We have
f AVw. Vw dx
1
= inf ~ C2 (~2, A) ~eVo
Ilwll2
where
AVw . Vw dx > / A o V w . Vw dx - eiiVwl[ 2. Hence,
1
f AoVw" Vw dx
C2(D,A ) > ( 1 - e c ~ -1) inf ~ -
Cl
-
-
cl
II ll
C~(~,Ao)
and
e
C 2 (gl, A) _ (1 + el
-
-
) C2(gl, Ao).
(6.8.19)
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
170 Now, we see that {
sup
AElgA
(1 + 3)D(Vv, y*) +
(1) 1+ ~
C2(f~' A) y, 12} 2 lldiv - fl <
feus
< (1 +/3) Do(Vv, y*) +
6
1 2(e c~-------------~ + Ily*II2§
6 sup f E(Vv - Aoly*) 9(Vv + Aoly *) dx) +
+ 2 Eeg
n
+ ( 1+
6 ) (l+13)C2(f~'A0, sup cl - 6 23 ~e~
f
Idiv
,,
- f o - 6 q o dx. (6.8.20)
n
Note that for any g E LZ(f~) sup f ( g - r ~o6.Y"
- {[gll2 + 2{Igl[+ 1.
t2
By this relation, we find the value of the very last term of (6.8.20). Recalling (6.8.19), we arrive at the estimate
emax
(,, T) <
< - -1 ( 1+ -cl
6 ) {inf ] 2 c l cl'j (1+]3) ( Do(Vv, y')+ ( e[--'l)y *2]2(6+ cl-e v'eQ" p>0
i(Vv - Aoly*) 9(Vv +
+ ~
Ao~y*)ld~)+
f~
6 ) (1+13)C2(~,A0) 2/~
+ ( 1 + ~Cl --6
(lldivy*-foll2 +2~lldivy*-foll+82)} 9 (6.s.21)
To represent the right-hand side of this estimate in a more transparent form,
6.8. INDETERMINACYINTHEDATA
171
we introduce the following quantities:
Moo(v, fl, y* ) := (1 + fl)Do(Vv, y*) 44- (14-~_1) C2(fl, Ao) ll divy, _ M ~ o ( v , ~ , V*)
8 /
5(
=
I(w - Ao
ly,
) ( w + Ao y*)l
+
f~
+
(1) C (n, Ao, f ldivv, - ,ol 1+~
cl-s
fl
Mol (v, fl, y*)
9= 6 ( 1 + ~ 1 ) C 2 ( ~ , A o ) l l d i v y . _ f o ,
M22(v, fl, y*)
C2(f~,AO)lldivy._
(,1)
Mll(v, fl, y*)
-
e
:=
(1 + ~) ~-i-
+
( 1 ) 1+?
8
Ily*ll 2
2(s + ~) +
clC2(f~,Ao) 62 2(c1-~)
Now, we see the structure of the upper bound
4(~, T) _-
1
C1
I
"
ee(v, T) in (6.8.7):
-
1 4-
el
inf -- 8
y* E Q *
/5>0
Mst (v, fl, y ) 4-/1//22(v,/3, y*)
.
(6.8.22)
s,t=l
The term Moo(V, fl, y*) does not contain the small parameters ~ and e. It coincides with the majorant constructed for the "mean" problem (A - Ao, I - /Co) and represents the major part of the approximation error. The terms Mlo, Mol, and Mll are given by some combinations of the weighted residual and small parameters ~ and ~. In principle, all these terms can be made arbitrarily small by taking v close enough to the exact solution u of the problem with A = A0 and l = f0 and y* close enough to AoVu. In contrast, the term M22(v, fl; y*) is always positive. This term contains the inherent part of the error, which does not depend on the accuracy of numerical approximations. Indeed, in all cases we have
01~2
M~2(v,P, y') _ eo "- O~(~, Ao)2(~ - ~)"
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
172
This quantity does not depend on the choice of v, /3, and y*. It gives an idea of the accuracy limit that could be achieved within the framework of the worst-case scenario. For example, if A0 is a unit matrix and fl = (0.1) x (0.1), then Cl 1 and C2(ft, Ao) = 1/27r 2. Assume that the coefficients are defined with the ambiguity rate c = 1% and l with 6 = 2%. In this case, 00 ~ 10 -5. -
-
"
A series of computable upper bounds is obtained if we replace Q* by a sequence of finite-dimensional subspaces {Q~} c Q*. Then, we have
e~(v, T) <_e2ke(V, T) -
1(1
= -C1
+
inf 01--~' y*EQ~ /~>0
}
Mst (v,/3, y* ) + M22 (v, fl, y* ) .
(6.8.23)
s,t=l
If Q~ c Q~+I, then the sequence {e~e(V, T)} monotonically decreases but may not tend to zero.
6.8.3
L o w e r b o u n d o f t h e error
To find a lower bound, we use the relation
I / AV(v
2
u) V(v
u) dx - sup ~4e(V,W), wEv0
fl
where Me(V,W)--/(~AVw.Vw+AVv.
Vw+.fw)dx.
f~ Recall that A~. ~ _ (c + c2)I~12. Then, by (6.8.4), we see that for any (A, f ) E ~4fA•162 and the respective solution ~, the following relation holds" 1
IIV(
-
)11
2
1
>- 2(r + c2) =
1 ~+c2
/ AV (v - ~). V (v - ft)dx = f~
sup Me(V, w). wEVo
6.8.
INDETERMINACY
173
IN THE DATA
Therefore, emin(V' T ) -
ainf e T ~1 IIV(v - a)l
1
>
12 >__ inf
sup
. M e ( V , w) >
-- 6 - } - C 2 (A,S)6blA xbll w6Vo
1
>
sup
inf
-- r -~- C2 w6Vo ( A , S ) 6 U A
M e (v, w) x/J'i
{-/ (1-
-_
~ A o V w . V w + A o V v . V w + low
r +1c2 -e-o'U'
)
dx+
n
+ inf
EEE ~6 Yz
( S( 1 -e
EVw.
Vw+EVv.
Vw
2
~
)
dx-6
S )} wqodx
(6.8.24) Here,
inf{-S(2EVw'Vw+EVv'Vw) dx} -
E6 E
fl
--li( 1 n
and
fw~dx}--Ilwll.
inf { -
~6.T"
n
Now, we obtain 2 emin(V, T) _>
>
1
sup { - i ( ~
-- 6-t-c2 wE Vo
)
AoVw'Vw+AoVv'Vw+fow
f~
-e
/I' l
I NVw + Vv | Vw[ dx
fl
-
allwll
dx-
}
9 (6.8.25)
174
C H A P T E R 6. L I N E A R ELLIPTIC P R O B L E M S
Introduce the quantities
moo(v,w)
=
-
/(1
-~AoVw . Vw + A o V v . V w + fow
)
dx,
n
-
,
mo,(w) =
-,~ll~ll.
Then, we represent the lower bound in the form
e~(v, T) :=
1
sup {mo0(v, w) + m01 (v, w) + mlo(w)} >_ 0.
(6.8.26)
wEVo
g + c 2
In this estimate, the term moo (v, w) contains the major part of the approximation error. It vanishes if v is a solution of the "mean" problem with A - Ao and l - fo. Two other terms reflect the influence of the small parameters 5 and e. To obtain a computable lower bound, we replace V0 by a finite-dimensional subspace Vok and solve a finite-dimensional problem ~ (v, T) >_ ~ e ( ~ emin
,
T) -
1
~up {moo(~,~) + mo~(,, ~) + m~o(~)}.
g + c2 wEVo~
The quantities eke2(v, T) show the efficiency of further computational efforts within the framework of the best-case scenario. If they are large, then approximation errors are significant, and using a more precise method of approximation is sensible. On the contrary, if all these quantities are small, then an approximate solution found is probably close to a function u E T, so that further mesh refinements do not give much additional information.
6.9
Error estimation
in t e r m s
of linear func-
tionals In this section, we apply two-sided error estimates given by ~ 4 e (or Me) and M e to deriving sharp bounds of errors evaluated in terms of a linear functional L Consider the problem (6.1.1) with I = f and another problem that is to find v E V0 + u0 such that
(AAv, Aw) + (~, w) = 0,
Vw E V0,
(6.9.1)
6.9. LINEAR FUNCTIONALS
175
where s is a given functional in Vo*. By Proposition 4.5.1, we have
(6.9.2)
where ~ is an approximation of u, ~ is an approximation of v, and E0 (~, ~) = (f, v-') - ( A A ~ , A~),
(6.9.3)
E~ (~, ~) = (AASf, ASt),
(6.9.4)
5f = ( u - ~), 5~ = ( v - ~).
Two-sided bounds of the term (AASI, A51) can be found by two-sided energy estimates as follows. Note that for any positive a we have 2(AASt, ASf) = 1 = Ill A (aSf + 15e) Ill2 - a 2 Ill A~f [[]2 _ ~ Ill A~e Ill2.
(6.9.5)
By the majorant M e (or M e ) and the minorant M e , we find two-sided estimates of the energy norms of 51 and 5t. Let y~, y~,/3f, fit, Wl, and wt be the functions and parameters that correspond to these estimates, i.e, 1
mY := .Me(~,ws) < ~ Ill A'b Ill~< .,~e(~,Z,,,y.;)"- M s, 1
m, . - .Me(~', w,) < E III ASl II1~< .Me(~',,~,,y;)"-
Me.
In this case, the quantity
III A(o,~ +
,Se)i11~-IIIA ( a u
1
+ -v-
1
a~
"6) lU~
(6.9.6)
can be viewed as the norm of the difference between the exact solution 1 u~l E Vo + (a + ~)uo of the problem
(g A u l l , Aw ) + (oL.f + l e, w) - O ,
Vw e Vo
(6.9.7)
and the function v~l - a ~ + ~v 1 - E Vo + (a + ~)uo, 1 which is an approximation of u~l. To obtain two-sided bounds of the norm (6.9.6) we use already known functions y~, Y'l, wf, and w~ and compute the numbers ~ I ~ = Me(,,?~, ~ I
+
1 ~), I
,
M ~ - M e (v~, ~, ~y'j + -y~ ). Ol
176
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
We note that ml, e = mfe(a), Mfe = M fe(a, ~) and positive numbers a and /3 can be taken arbitrary. By (6.9.5), we obtain (AhSf,hSe) < -
me a2mf - ~-~-} := ff)~$,
inf { M f e a,flER+
(Ai(f/,, iSe) >_ sup { m f e - a 2 M / -
Me
--~-} := ff)~o.
aER+
Recalling (6.9.2), we now deduce a two-sided estimate
+ E0( ,vD <
( e, u -
<
+ Eo (
v-) ,
(6.9.8)
Note that if y~, y~, wf, and we provide accurate two-sided estimates of the energy error norms in the problems (6.1.1) and (6.9.1), then Mf e and m f e furnish accurate two-sided estimates in the problem (6.9.7) and, consequently, (6.9.8) yields sharp upper and lower bounds of (g, u - u/.
6.10
Practical
implementation
In this section, we analyze the effect of duality error majorants (DEM) and compare them with the results obtained by other methods. 6.10.1
General
scheme
Two-sided estimates obtained in Section 6.2 suggest a general algorithm for solving Problem P with any a priori given accuracy A. First we choose a set of finite-dimensional spaces {Irk } C V (dimVk = Ark ~ +
I Choose a desired tolerance A, set k = 1, s = 1, and define Vk and
Yr. S t e p 2 Find Uk by solving Problem ~ on the space Vk. Step
3
Find M~ s
-
inf M~ (Uk,/3, y*). y'EYe*
6.10. P R A C T I C A L I M P L E M E N T A T I O N
177
S t e p 4 If M~ 8 < A, then go to Step 8. S t e p 5 Find Ad~ -
sup .h4o(Uk,W). wEv~+l
S t e p 6 If Ad~ > A, then replace k by k + 1, Vk by a refined space Vk+l and go to Step 2. S t e p 7 Replace s by s + 1, Ys* by a refined space Ys*+l and go to Step 3. S t e p 8 Print Uk and stop. The first step defines spaces originally used for obtaining an approximate solution (Step 2) and finding the first (rough) upper estimate of the error (Step 3). If A exceeds the majorant (Step 4), then the desired accuracy is achieved and the process passes to the final Step 8. If it is stated that the minorant exceeds A, then Uk does not have the required accuracy and it is necessary to solve Problem 7~ on a refined subspace Vk+l (Steps 6 and 2). The situation that occurs if
deserves a special comment. In this case, it remains unclear whether or not Uk is close enough to u. To answer this question additional efforts are required. One possibility is to refine Vk and another is to refine Ys*, trying either to find a better approximation or to compute a better estimate for the existing approximation. Since uk -+ u and our two-sided estimates are exact (see Section 3) , this theoretical algorithm will end with finding a proper approximation Uk. However, any particular computer has a certain limited power, so that any problem can be solved only if A _> Ao, where A0 depends both on the problem considered and the computer used. Certainly, the above algorithm is rather schematic and can be viewed only as a skeleton of reliable numerical algorithms to be used in practice. Such algorithms should include numerous improvements focused first of all on accelerating the process. It could happen that these computations are terminated by time limitations. In this case, the very last value of M~ l shows the best accuracy achieved, which gives an idea of the required power of a computer to be used for finding an approximate solution with the desired tolerance A. Finally, it seems worthwhile to add one more remark. Steps 3 and 5 are related to variational problems, so that one may ask about the sensitivity of
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
178
the algorithm with respect to inaccuracy of their solutions. To clarify this point, we recall that M e ( v k , ~ , y*) provides an upper bound of the error for any y* E Y* and Me(vk , w) provides a lower bound for any w E V. For this reason, exact solutions of these variational are, in general, not required. For example, it may occur that on some stage of the minimization procedure the value of M e becomes less than A. Then computations may be terminated regardless of that how close "current" function y* is to the minimizer y*. 6.10.2
Minimization
of the majorant
Consider the majorant M e ( v , ~ , y * ) defined by the relation (6.3.15). A coarse upper bound of the error is easy to compute by setting y* - GAy, where G is an operator that maps Av to Q* (e.g., if v is a finite element approximation, then G is the standard gradient averaging operator). Then, we obtain the error majorant
M e ( v , 3 ,GAy) = (1 + 3)D(Av, G A v ) +
( 1 ) 1+ ~
c2 ,2 -~ll/+ A*GAvl .
A sharper estimate requires a minimization of M e with respect to the variables y* and /3, which can be done by a direct minimization of M e or by finding a minimiser as a solution of the respective the system of linear simultaneous equations. The latter way is considered below. Assume that
v;, ..., v;),
v* e Y : .=
where y~* E Y* are given trial functions. Then any y* can be represented in the form
k (6.10.1)
j--1 Define the matrix M ~ with entries
* m~t - ( 1 + ~)
C2
(A-ly~,yt) + ~(A*y*,h*y;)
and the vector F ~ with components F ] = (1 +fl)
( (Av, y ; ) - ~(A*y~,/) c2 ) .
/ ,
s,t = 1,/
179
6.10. P R A C T I C A L I M P L E M E N T A T I O N Then
Mr
= (~(~,7) "- -~M 7 "7-- F~ "7 + #~,
where _ (1 + j3)
C2
12
and 7 = {%}In view of the property of M s , we have 1
III A(v - u)
III
inf 0(/3,7). BeR+, ~Y:
(6.10.2)
The problem on the right-hand side in (6.10.2) can be solved with help of a direct minimization method that creates a sequence {/3j,7j} as follows: (A) Find f/j E ~ such that (I)(/3j, 7j) = inf r
7j) := MS
and (B) Find 7j+l E Ys* such that ~(/3j, 7j+1) = inf (h(f~j,7). 7~Y: This algorithm creates a decreasing sequence of numbers M S tending to an upper bound of the error. Each number is a reliable upper bound of the error, so that if MS _ A, then the process may be terminated. Remark 6.10.1. Minimization of the majorant can be performed by the least-square methods developed for some classes of elliptic problems (see, e.g., [33]). Also, we note that the minimization problem can be partitioned into several subproblems associated with some blocks of 7. Therefore, one can accelerate the minimization process by various parallelization procedures. Remark 6.10.2. Let the functions y~ be properly chosen and M E has an inverse matrix for any positive/~. Then, (~(/~,7) is minimized by the vector
and the respective minimal value of this functional is easy to find. In this case, for a given f~ we obtain the estimate 1 Ill h ( , -
_
Ill <
#a
-
1M
1Fa 9 Fa
(6.10.3)
180
6.10.3
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
Effectivity index and
shape
index
In the examples considered below, we evaluate the quality of error estimates by means of two numbers. The first number is the so-called effectivity index /eft :=
Me e
Me - e = 1+ ~ , e
(6.10.4)
where 1 -
Ill h (, -
III
and M e is an estimate of this quantity obtained by an error majorant. If /eft is close to 1, then M e gives a sharp upper bound. The second number characterizes the distribution of subdomain errors. Let 7" be a partition of ~ into m nonoverlapping subdomains ~i, i = 1,2, ...m. For obvious reasons, it is desirable to have an error majorant that not only shows the total value of the error, but also indicates errors in various subdomains. We suggest to evaluate the latter capacity by a special number called the error shape index m
i esh -- 1 + i=1
>__ 1.
(6.10.5)
e
In the above relation, ei is an error associated with ~i and #i denotes an approximate value of this local error computed by the majorant. If i esh is close to 1 then the majorant provides a sharp error estimate and, in addition, the corresponding numbers #i serve as good indicators of subdomain errors. Now we turn to examples.
6.10.4
N o n - G a l e r k i n approximations
Below, we present results of several numerical tests that demonstrate the capability of the D EM error estimation method to correctly estimate errors of approximate solutions regardless of their closeness to the exact one. We begin with a simple 1-D problem (c~(x)u')' = f(x), = 0,
(6.10.6) =
(6.10.7)
It is easy to see that (6.10.6) is the Euler equation for the functional b
a
6.10. P R A C T I C A L I M P L E M E N T A T I O N
181
Assume that
a E L~176
a >_ cl > O,
f C La(a,b).
(6.10.8)
Then, the problem (6.10.6)-(6.10.7) is uniquely solvable. To estimate the errors, we apply the general scheme with U - L 2(a,b),
Y * - L 2(a,b),
o
V0 - H 1(a, b),
Q* - H 1(a, b),
and
Vo + uo - {v e H 1(a, b) I v(a) - O, v(b) = Ub}. In this simple case, A is the differentiation operator A'y* - v* E H -1, where
/
b
y*r
= -
a
/
o
b
Vr ~ H ~ ( a , b ) .
v*r
a
Now, the duality error majorant Mm has the form (cf. (6.3.15)) b
Mm(v,~,y*) - 1 +2~ f l a y ' - y *
[2 dx+
a
b
+ I -;/3 c2(~,b) /[Y*' - f[2 dx.
(6.10.9)
and the minorant M e has the form b
,~(v,
w)=-
/(1
~lw'l = + ~v'w' + / w
)
.
(6.10.10)
a
The exact solution admits the integral representation
u(x) -
/ 1 / ~(t) a
x f (z)dzdt nu -~ a
Ub -
~ a
1
f (z)dzdt
. (6.10.11)
a
In the tests presented below, we consider piecewise affine approximations v computed on regular meshes with N intervals. Our aim is to show that M e and M e provide good error estimates for exact solutions of the respective
182
CHAPTER
6. L I N E A R E L L I P T I C P R O B L E M S
finite-dimensional problems as well as for solutions that do not satisfy the Galerkin orthogonality condition. Therefore, we compute approximations by a direct minimization of J and analyze errors arising on different stages of this process. To find a sharp upper bound, we minimize M s with respect to y* and ~ starting from the function y~ = G(v'). Here, G : L ~ ( a , b) --+ H 1(a, b) is a simple averaging operator. For example, G may be defined by the relations =
1
- 0) + v'
=
+ 0)),
-
Computing inf M s ( v , ~, y~ ), ~>0 we find the first (rough) upper bound of the error. It is further improved by minimizing M s with respect to y*. To compute a lower bound, we define a recovery operator R that maps Y* to Vo + u0: R y * ( x ) :=
a(t) dt + -~
a(t) dt
Ub-
.
It is easy to see that Ry* = u if y* = au'. The quantity M e ( v , R y * - v)
gives a lower bound of the deviation. Let us begin with an elementary example in which a ( x ) = 1, f ( x ) = 2, a - O, b - 1, Ub -- 1, and C(a,b) -- 1 / r 2. In this case, it is easy to find the exact solution u-~
+
1-
x
and compare the actual errors with their estimates. Below we do this for various approximations v. First we take a "very rough" approximation v = x (for c - 0.5 the functions u and v are depicted in Fig. 6.10.1(a). In this simplest case,
1 ~ll(v - u)'l ]2 - ~o 1(1 - 2x)2dx - c2/48 ~ 0.006c 2.
6.10. P R A C T I C A L
IMPLEMENTATION
1.2
183
0.004
n
I
Errors DEM
V U
1
i
0.8
i
.I 0.002
0.6
al~
0.4
a
a a '1
0.2 0
0 0
0.5
1
la
ill% ~a lilil Ill lieD I Ill llll~ ~ll mallei Ilia mmlmmh ~mlm iililll lilli iaalae7 ~lani illlllm~ HNDIII Dllllll ~ IIIIn lmllllll Illlllil mmmmmlmmm I mmmmmmmmm
0
(a)
0.5
1
(b)
Figure 6.10.1: (a) Exact solution and an approximation, (b) distribution of local errors.
For any partition of the interval (0,1), the function v - x can be viewed as a piecewise affine approximation of u. The corresponding subinterval errors e2i for the uniform partition with 20 subintervals are depicted in Fig. 6.10.lb. We compare t h e m with the errors computed by the m a j o r a n t (6.10.9) with different y*. If we set y* = y~ - 1 (cf. (6.7.4)), then the first term vanishes and M e (v, f~, y~') -+ c 2/27r 2 ~ 0.05 c 2
as fl --+ +oc.
Hence, we see t h a t the quantity obtained by the only one (residual) term of M S gives a rough upper bound. However, if we take y* = = c x + 1 - c/2, then the second term vanishes and we obtain another upper estimate: 1 C2
M . ( v , fl , y ~ ) -~ -~
x 0
dx--~
a s f ~ - + O,
184
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
which is exact. Thus, we see that the value of a computed upper bound strongly depends on the function y*. Now let us show that this dependence also takes place when the majorant is used as an indicator of local errors. Namely, if y* is defined in the process of minimization of M e , then the local errors given by the majorant practically coincide with the true distribution (typical picture is presented in Fig. 6.10.1(b). However, attempts to use simplified forms of the majorant that eliminate either the first or the second term may lead to quite different results. In our simple case, this fact is easy to demonstrate. Indeed, the choice y* - y~ - 1 zeroises the first term, so that b
1 +/3c2 / [2 M e (v,/3, y~) 2~ (a,b) If dx. a
In the integral sense, this majorant gives a corect result, but cannot serve as an error indicator, because its integrand given by the residual term is constant (f = 2), while the real error distribution demonstrates strong heterogeneity. It is easy to present even a more striking example showing that the use of averaged derivatives without taking into account the second (residual) term of M e may lead to a confusion. In our case, all sensible averagings of v ~ = 1 give exactly the same function, so that G(v') = G(1) = 1. Therefore, G(v')
-
v'
-
0
and the estimator I[G(v') - v'l[ (which is, in fact, the first term of M e for y* = G(v~)) leads to a wrong conclusion that the error is equal to zero. However, taking into account the second term changes the situation and gives a correct upper bound of the error. This observation shows that error indicators based on comparing the gradients of approximate solutions with their averaged values are not applicable to arbitrary conforming approximations without the second (residual) term. To give further illustrations, we consider the functions u~ = u + ~r where ~ is a number and r is a certain function (perturbation). Approximate solutions (whose errors are measured) are piecewise affine continuous interpolants of u~ defined on a uniform mesh with 20 subintervals. If ~ = 0, then v coincides with the interpolant of u. In the first series of tests, we take r = x sin(zrx) and ~ = 0.1, 0.01, 0.001, and 0. The results are presented in Table 6.10.1. For r = x sin(27rx), the results are exposed in Table 6.10.2.
6.10. P R A C T I C A L I M P L E M E N T A T I O N
185
0.008
Errors DEM
Errors GA
0.006
0.004
0.002
~._+L[
0 0
0.5
0.5
(~)
(b) 1
v!
-
U
0.8
I
I
I
left
-
-
I
~ .
.
.
.
.
.
.
.
.
, . . . . . . .
, . . . . . . . . . . . . . . .
,-
..=
0.6 0.4
-
,,7
,'7
0.2 -
=
I
0.5
i,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
m
I,
,7 o
I
0 0
0.5 (c)
1
I
I
1 3 5 7 9 (d)
Figure 6.10.2" Error estimation for ~ - 0.1.
I
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
186
I
I
I
Errors DEM
Errors GA . . . . . . .
8e-05
! "
I'1
#"" Ii1!!
4e-05
ji,~
Im
IIii
im
"'
0 0
0.5
1
0.5
(a)
(b) I
~
3
1
I
I
left
I
I
----:--
_
1.5 8e-05
_ Error /
,
7" ,
, , ~
4e-05
~
0.5 ~ %
.
,
~
~
~
,
, ,
, ~
, ,
~
,
,
.=..,.,,
,m
,m
0 0.5
(c)
1 3 5 7 9
(d)
Figure 6.10.3" Error estimation for 5 - 0.01.
,,...
,=b
.,.
6.10. P R A C T I C A L IMPLEMENTATION
187
I
I
Errors DEM
Errors GA .
4e-05
.
.
.
.
.
m
2e-05
0
I
i 0.5
0
0.5
1
(a)
(b) I
g
. . . . . . .
10 9~
4e-05
.
.
~
L.
......
!
I
................ , L~S~ ~ .~S S uii ,IDIjlIIII
i,]1
9 . . . . . . . . .
14,1 iii,
beta . . . . . . . ID u I
1.5
[
-~
_
1 2e-05
O5
0
0 0.5 (c)
1
3 5 7 9 (d)
Figure 6.10.4: E r r o r estimation for 5 - 0.001.
188
C H A P T E R 6. L I N E A R ELLIPTIC P R O B L E M S
, -
"
I
" -
,,,
,,
I
Errors DEM
4e-05
,, i,
-
Errors GA
- -
............. .......
iii
...... :11' 1 I!
3e-05 2e-05 le-05
J
i
0
0
0.5
0.5
1
(a)
(b) ! .,,,,,.,m.,.,.,,..m...,,,..
10 -
beta .......
................
20
-
-
-
1.5
m ~ ~
-~m
~
m
.
.
.
.
.
.
.
4e-05
.
.
.
~.
1
3e-05 0.5
2e-05 le-05 I
0
0.5 (c)
0 1 3 5 7 9 (d)
Figure 6.10.5: Error estimation for ~ = 0.
189
6.10. P R A C T I C A L I M P L E M E N T A T I O N
0.02
I
I
Errors GA . . . . . . .
Errors DEM 0.015
0.01
0.005 - ~
I Ill
IL
, i'r4.-_,
0
0
I
0.5
0.5 (a)
0.8
(b)
~
'v ! ---'-- ] u !,
/,/ Z''''
0.6 0.4
I
I
beta
.......
1
I
0
I
I
leff
................!i
0.2 t t 0 ~'t 0 0.250.50.75
I
1.5
., .......
0.5
(c)
'
, .......
, .......
i!
-
i ..........
0
~,~
'.
~
,
I
I
1
3 5 7 9 (d)
Figure 6.10.6: Error estimation for ~ - 0.1.
t
t
190
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
I
Errors
'
I
I
I
Errors GA . . . . . . .
0.0002 -
0.0001
i inll !,11
0 0
0.5
1
0.250.50.75
(a)
(b) I
I
0.0002
10 ................ 20 _Error
I
I
1.5 i
.,. ......................
; ........
I
0.0001
0
l,lllll "=" =. ~'~fi
. m . ,
.
.
.
.
.
.
.
~.
.
.
.
.
.
.
, .
.
.
.
.
.
.
.
.
.
.
.
,-
o
I I I~
0.5 (c)
,==
,
0.5 S~
I
left beta..-------...,._
0
,,,,. .......
I
1
J~ o~..,,p .~, ,~, .~..~, .~, --. -.- -.,,, -.i,~
I
I
3 5 7 9 (d)
Figure 6.10.7: Error estimation for ~ - 0.01.
I
6.10. PRACTICAL IMPLEMENTATION
191
Errors
Errors DEM
GA
----,W j t m
4e-05
ItW flmm I I
u
3e-05 2e-05
IlU
le-05 I
0 0
0.5
0.5
(a)
(b)
2 ...,
m
,..,..,,.,
m
i
m
10 ................ 20 Error
beta . . . . . . . 1.5 1
4e-05 3e-05
0.5
2e-05 le-05
0
0 0.5 (c)
1
3 5 7 9 (d)
Figure 6.10.8: Error estimation for 6 = 0.0001.
192
C H A P T E R 6. LINEAR ELLIPTIC P R O B L E M S
0.1 0.01 0.001 0
e 0.019692 0.001022 0.000835 0.000833
Table 6.10.1: 2M~ 2Me 0.019743 0.019683 0.001025 0.001013 0.000839 0.000827 0.000836 0.000825
left 1.003 1.003 1.005 1.004
iesh 1.018 1.011 1.002 1.002
0.1 0.01 0.001 0
e 0.068365 0.001509 0.000840 0.000833
Table 6.10.2: 2M~ 2.~ G 0.068543 0.068357 0.001520 0.001500 0.000844 0.000832 0.000836 0.000825
ieff 1.003 1.003 1.005 1.004
iesh 1.045 1.042 1.004 1.002
Moreover, we compare the actual values of subinterval errors with the indicators given by the gradient averaging (GA) and duality error majorant (DEM) methods. In the first case, our indicator is a function proportional to ( v ' - G ( v l ) ) 2, where G is an averaging operator that defines nodal values as a simple average of two neighbor values. In the pictures, the values of this indicator are properly scaled in order to fit the actual value of the error. The second indicator is simply the integrand of the majorant M s. Four pictures presented in Fig. 6.10.2 refer to the first line of Table 6.10.1. Picture (c) depicts u and v. True distributions of local errors are shown in (a) and (b) by impulses and compared with the DEM and GA indicators. Picture (d) gives an idea of how the values of the majorant and the parameter ~ change in the process of minimization. Here, the values of the x-axis are the iteration numbers (or time units). In Fig. 6.10.3 these numbers also appear in the picture (c), where they show how the integrand of M s converges to the true error in the process of minimization. Subsequent pictures are associated with other lines of the above tables. We see that, irrespective of the actual appearance of the approximate solution v, the majorant M s yields an accurate estimate of the global norm of the error and also rather accurately reproduces the distribution of local errors. Also, we observe that GA indicator provides a good error indication for small values of ~. Also, we tested the DEM error estimation method for various twodimensional problems. One typical example is presented below. Let u be a
6.10.
PRA CTICAL
IMPLEMENTATION
Table 6.10.3: Estimate left 1.0812 1.477 0.9791 1.334 0.8069 1.102 0.7418 1.013
N 1 3 6 10
193
iesh
2.184 1.790 1.215 1.027
solution of the problem Au = 4x 2 in fl e (0, 1) • (0, 1), 1 lx4
uly=o - ~-
+ x,
1 4
ul,,=~ - ~-~ + x + 1.
1 4 + x + y. We compare it The exact solution of this problem is u = ~x with the approximate one v = u + xysin(Trx)sin(Try*). To find the best upper bound for the error e, we minimize the majorant on the spaces Ys* constructed by means of the polynomial trial functions x i y j, it j = O, 1, 2 .... Fig. 6.10.9(a) shows the distribution of local errors and the errors computed by minimization of the majorant on Y* with 3, 6, and 10 trial functions are depicted in Figs. 6.10.9(b), 6.10 .10 (a) , and 6.10.10(b), respectively. The numbers are given in Table 6.10.3. We see that the DEM furnishes a good estimate of the energy norm of the error and of its shape even for N = 6. If N = 10, then the norm and the shape of the error are found with a high accuracy. Certainly, in more complicated examples it may be necessary to invoke more trial functions in order to accurately estimate an error and its distribution. Moreover, if gl has a complicated structure, then "global type" trial functions that we used may be inconvenient. In such a case, it is better to construct Y,* by local functions usually used in finite element methods.
6.10.5
Duality error proximations
majorants
for finite
element
ap-
Now, we focus our attention on finite element approximations of 2-D boundaryvalue problems. We consider problems with homogeneous boundary conditions in domains with polygonal boundaries. Results obtained by the DEM method are compared with other three error indicators widely used for finite element approximations. They are as follows: 9 the indicator ~72 - I I A V u h
- G(AVuh)ll 2
194
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
Figure 6.10.10" Integrands of the majorant for (a) N - 6 and (b) N - 10.
6.10. PRACTICAL IMPLEMENTATION
195
where G(AVuh) is a piecewise affine function defined at the central point xi of a patch 12i by the relation (cf. (6.4.6))
IT,ll
G(AVuh)[x,
(AVuh)[T~.
9 the indicator 72 based on the L2-projection of AVuh onto the space Vh of piecewise affine vector-valued functions (cf. 6.4.10): 72
-
inf
y* E Vh
2 IIAVuh-y*ll~.
9 the indicator _
IEI
IIj(aVuh" v)ll 2,
E6~h
where ~h is the set of interior edges and u is the unit vector normal to the edge E. The error indicators 71 and 72 were considered in Chapter 4. It is easy to see that 773 coincides with the part of the explicit residual error estimator that contains jumps. It is known (see, e.g., [46]) that for low order finite element approximations usually edge terms dominate in the residual type estimates and, therefore, can be used as an error indicator. We compare the results obtained by these indicators with those computed by D EM, paying attention to (a) the accuracy of the energy error estimate and (b) the quality of the local error indication. It should be noted that in many cases (especially for problems with smooth data), all indicators demonstrate close results. However, in other cases results are different. Below, we present several examples taken from [77], which show that various situations may occur.
In the first pair of examples, we consider two cases that differ only by coefficients of the matrix A. As we will see, the "standard" indicators give good results in the first case and are much less reliable in the second one. The error estimates computed by the DEM technology are quite reliable in both cases.
Example 6.10.1. Let us consider the equation div AVu + f = 0
C H A P T E R 6. LINEAR ELLIPTIC PROBLEMS
196
in a square domain f~ E ~2 and take f- 1 f = 0
forx E (0.5, 1.5), otherwise,
all = 10 all - 1
forx E (0.5, 1.5), otherwise,
a12 - a21 - 0 , a22 - 1. In this example, all the tests are computed for a certain fixed partition of the domain. To visualize the error indication, we mark elements by two colors (see Fig. 6.10.11): dark (if the error on a selected element is less than the mean value computed for the whole domain) and light (in the opposite case). This information can be further used in the simplest mesh-adaptation procedure that refines elements of the second group. The quantity p shows the number of correctly classified elements (in percent). The results of error estimation are compared with the actual values of errors obtained by using exact solutions. If in an example considered the exact solution is unknown, then the so-called "reference" solution is used instead. Such a solution is rigorously computed on a mesh that is much finer than those used in the error estimation tests. In all the examples, the mesh used for reference solutions has eight times more degrees of freedom than a primal mesh. Numerical results are collected in Table 6.10.4. In this example,
Ill
UI'-iii
-
ill- 0.02578
and the normalized value of the error is
ili eh ill = 5.3%. ill iii The respective results for the effectivity index are depicted in Fig. 6.10.12. From the values of p in Table 6.10.4 and the diagrams in Figs. 6.10.11 and
estimate I p
Table 6.10.4: r/1 ~2 0.03060 0.02952 1.19 1.15 90.2 90.3
r/3 93.0
v/M 0.02656 1.03 98.9
6.10.12, we conclude that the DEM gives an accurate error indication and a sharp upper bound of the error norm (p - 98.9% and I = 1.03). The
6.10. PRACTICAL IMPLEMENTATION
197
behavior of the effectivity index depicted in Fig. 6.10.12 shows that one time unit is enough for obtaining a sharp error estimate. It is easy to see that the majorant can also be used as an effective indicator of local errors. We present the true distribution of these errors and their estimates computed by the majorant as 3D graphs that present their averaged values on each element (see 6.10.12). By comparing these two pictures, we observe that the majorant accurately shows the distribution of local errors. The indicators ~1-~3 are also quite good. Among them, ~3 (P = 93.0%) is the best (but this indicator does not give a guaranteed upper bound of the total error). However, the quality of these estimates is lower than of those made by the D EM method.
Example 6.10.2. In the second example, we take the same data as in Example 1, but define the coefficient all as follows: all - 1 a~ 1 = 10
forx E (0.5, 1.5), otherwise.
In this case, the energy norm of the error is 0.02608. The respective results are collected in Table 6.10.5 and Figs. 6.10.13 and 6.10.14. It is clear that here, as in the previous example, the DEM technology provides a high accuracy of the error estimation. It was observed that, in the minimization
estimate I p
Table 6.10.5: r]l r/2 0.03346 0.03211 1.28 1.23 61.1 62.5
r]3 63.3
v/M 0.02680 1.03 97.9
process, the second term of the majorant diminishes with respect to the first one. As in the previous example, finding a sharp energy estimate takes about two time units. In contrast to the DEM, all indicators give an inaccurate error distribution (see Fig. 6.10.13), but ~1 and ~2 provide quite good estimates of the global norm. The above results show that, in some situations, the indicators r/l, ~2, and ~3 may work quite good, but in another one their quality may be not good enough to guarantee the most effective mesh-refinement. Since it is difficult to predict a priori which case we are dealing with, the quality of such error indication may be not optimal. The DEM provides quite reliable error estimates in all cases, but the determinition of a sharp distribution of local errors may require an extra CPU-time. We also remark that the
198
C H A P T E R 6. LINEAR ELLIPTIC PROBLEMS
Figure 6.10.11"
6.10. P R A C T I C A L IMPLEMENTATION
Figure 6.10.12:
199
200
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
Figure 6.10.13"
6.10. P R A C T I C A L IMPLEMENTATION
Figure 6.10.14:
201
202
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
indicators give some values that are difficult to improve. In contrast, the quality of the DEM can be increased up to any required level (it is only the matter of time). In [77, 181], the reader can find a more detailed discussion of these questions and results of various numerical tests, in which the known a posteriori error estimation methods for finite element approximations were compared with those computed by the duality error majorants. At the end of this part, we make the following note. Since majorants contain constants depending on the respective domain fl, we must either find them analytically or compute their upper bounds by a certain numerical procedure. For problems with Dirichl~t boundary conditions such a procedure is easy to construct. In other cases, this task my require a more complicated analysis (see [145]). Finding such global constants is an important task that requires a deeper investigation.
6.10.6
Computational c o s t s
Let us estimate computational expenditures required to compute an upper error bound by means of the duality error majorant. Assume that f~ is a square domain and a finite element sampling is given by a regular (n • n) mesh with 6 elements in a patch. For large n we can estimate the total amount of elements N by the quantity 2n 2 and the number of edges M by 3n 2. In this case, the computation of the majorant (l+/3)l]Vu h
--
y*l12,• + ( 1 + ~ ) C ~ , , d i v y * + f , , 22 , f l
requires the following steps: (a) compute Vuh on each simplex T (which amounts to approximately N simple computations); (b) find the averaged gradient y* = G(Vuh) (the respective expenditures are proportional to the number of nodes and require only simple summation operations); (c) find y* - Vuh for any simplex T (approximately N simple computations); (d) find div Vuh for any simplex T (approximately N simple computations); (e) find one global constant C~. This step may not lead to extra computations provided that C~ (or an upper estimate of it) is defined
6.10. P R A C T I C A L I M P L E M E N T A T I O N
203
analytically. Otherwise, it leads to a single complicated computation of a constant. However, it is important that this constant does not depend on the mesh, so that in a process of mesh refinement no further computations of this constant are necessary; (f) compute N + N volume integrals and find M s (2N simple computations). Certainly, the upper bound of the error computed in such a simple way may be coarse. However, changing y* by some minimization procedure we can rapidly decrease the value of the index up to the numbers about 1.5 or 2. Further improvements of the effectivity index usually requires more time. We may compare these expenditures with the costs required to compute the explicit residual type estimate N
M
c2ijllj(vij 9 VUh)ll 22, E i j
cx,II div Vuh + fll~,T, + i=1
"
i,j=l (i>j)
In this case, we need to perform the following steps: (a ~) compute Vuh on each Ti (N simple computations); (b ~) compute normal vectors vii on each edge Eij (M simple computations, which, however, must be repeated upon changing a mesh); (c ~) find vii 9VUh for any simplex T (approximately 3N simple computations); (d ~) find div Vuh for each simplex T (N simple computations); (e ~) find the constants cli and c2ij ( N + M complicated computations, which must be repeated upon changing a mesh); (f) compute N volume integrals and M line integrals (N + M simple computations). It is not difficult to see that (a) and (a') are identical, while (b), (c), (b'), and (d) are not expensive and lead to comparable expenditures. The steps (d) and (d') are identical. However, (e) is much simpler than (e'), because we need to estimate only one global constant. In (d), we define local constants that depend on the mesh and must be recomputed after each mesh refinement. Even if such a recomputation is based upon some a priori computations made for a certain referenced element (or elements), the determination of all constants must take into account new geometric data
204
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
(angles, areas, etc) and would not be cheap. Also, a change in the degrees of polynomials forming the approximation space may lead to an expensive recomputation of all interpolation constants. Finally, we observe that (f) and (f) lead to approximately the same expenditures. Thus, we conclude that expenditures required for computing the majorant for a certain y* (e.g., for y* = G(AVuh)) does not exceed those that must be spend for the standard residual based error estimator. 6.10.7
Conclusion
Functional type a posteriori error estimates are universal error estimation tools that can adequately deal with conforming approximations of all types. Its combination with an effective numerical method for solving Problem P can give a reliable numerical strategy able to find approximate solutions with any a priori given accuracy. If an approximation v is close to the exact solution, then the DEM method often gives a good upper estimate even on the first step (e.g., without any minimization of M$ with respect to y*). In this case, we define y* by a simple averaging procedure and find an optimal parameter ~ in Mr y*, ~), which is a simple and easily solved task. In general, finding a sharp error estimate requires solving an additional minimization problem whose difficulty is comparable with the difficulty of the original problem. This effectively means that real expenditures for finding approximate solutions with a guaranteed accuracy can noticeably exceed those required for finding a solution without verifying its accuracy. We believe that, in general, such a "reliability fee" is unavoidable and one cannot obtain a posteriori error estimates that are simultaneously universal, exact, and computationally cheap.
6.11
Comments
At the end of this section, we prove one useful result that justifies the asymptotic behavior of the majorant Me usually observed in numerical experiments. P r o p o s i t i o n 6.11.1. Let (~k, Y~) E I~_ x Q* be a minimizing sequence for the majorant Mr ~, y*) and f~k ~ O.
(6.11.1)
Then the first term of the majorant tends to the energy norm o] the error and the second one to zero.
6.11. COMMENTS
205
Proof. Since (/3k, y~) is a minimizing sequence, we have M$(v,~k,Yk) - (1 + 3 k ) D ( A v , Yk)+
1 +-~k
-2 lie + A*y;ll 2
where C is a positive constant independent of k. bounded, so that for all k we have f~k _/3. Hence,
III Yk, [112.<- C
and
c,
The sequence f~k is
1 A* p, 2 ~k-kll (Y~ -- )il --< C.
(6.11.2)
From this fact, we conclude that there exists a subsequence (for the sake of simplicity we keep for it the same notation y~) such that y~ -~ y*
in Y*,
(6.11.3)
A'y; -+ A'p*
in U.
(6.11.4)
Therefore, lim M e (v, 3k, Y~) ___
m--+ r
> lim D(v,y~r > - m--,oo - 51 111Av III2 + ~1 [HY* 1112. -- (Y* , Av).
(6.11.5)
Recall that (3k, Y~) is a minimizing sequence for Me, so that ~lim _ ~ M e (v, f~k, y~) - ~1 OilA ( v - u)ill ~ and, therefore, 1
1 1 y, 2 ll1 h (v - u) Ill2 _> 2 111Av Oi2l + ~ IJi Oil. - (y*, hv).
This inequality leads to another one 1 p. _ 1 y* 2 - (p*, hv) + ~ [l[ 1112> - (Y*, hv) + ~ Ill Ill,.
(6.11.6)
Note that lim (y; - p*, Aw) + (p* - y*, Aw) - 0,
k--++c~
Vw e Vo.
In view of (6.11.4), the limit on the left-hand side is zero, and we find that (y*, Aw) - (p*, Aw),
Vw E Vo.
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
206
Take w = v - uo E Vo and use the relation (y* - p*, Av) - (y* - p*, Auo) + ( y* - p*, A(v - u0) ) - (y* - V*, Auo). Then, (6.11.6) yields 1
p,
(p*, Auo) - ~ [[1
_
1
[112,< (y,, Auo) - ~ 1[[y 9 1112,.
(6.11.7)
Since p* is a maximizer of Problem ?*, we have
I*(p*) >_I*(y*) which is equivalent to 1
p,
(p*, Auo) - ~ 111
2
1
11[,_> (y*, Auo) - ~ [[[
y,
[112,
.
(6.11.8)
Now (6.11.7) and (6.11.8) leads to the equality 1
p,
(p*, Av) - ~ 111
2 1 y, 2 [11,- (y*, h v ) - ~ Ill 11[,.
(6.11.9)
However, Problem P*, has only one maximizer, so that (6.11.9) means that y* = p*. Now, we obtain 1
Ill A ( v - u)1112- m-~oolimMr
#1 + #2,
(6.11.10)
where #1 #2
-
-
lim (1 + ~k)D(Av, y;),
k--+oo
lim
1+
2- Ill + A*p~ll ->" 0
We know that y~ weakly converges to p* in Y*. Therefore (we use lower semicontinuity of the norm), 1
l im (~ I[] A v ill2 + ~ []] y~ ]]]2,-(AV, Yk))>_
k--+oo
1 1 p, 2 1 - 5 [li hv Ill2 + ~ [1[ lU, - (hv, p*) - ~ ][I h (v - u) Ill2
6.12. NOTES FOR THE CHAPTER
207
and from (6.11.10) we conclude that #1 - ~1 ][[ A(v-u)I[[ 2 and #2 - 0. Now, we have 1 1 p. 2 , ) #~ - ~ I[I Av I[[2 + ~ [[[ [[[, - (Av p* =
. + Z~ D (Av, y~) .) -~oo 2 ]Uhv [[[2 + 51 [[[ yk. [][2._ (Av, yk)
lim ( 1
=
1
-- -2 j[[ Av I[[2 - ( h v ' p*) + kl im 51 IIIYk, 1112,9
Thus, ][[ y~ U[.-~lll p* ][Ik and, consequently, y~ strongly convergence to p*. In view of (6.11.4), we conclude that y~ strongly converges to p* in Q*. [:]
Remark 6.11.1. Such a behavior of the majorant (when the first term tends to the energy norm of the error and the second one rapidly decreases) was really observed in the vast majority of numerical tests. Some of them are presented in [77, 78, 173, 181]. Remark 6.11.2. Consider linear elliptic equations of the second order. In this case, Q* - H(div, f~). Assume that y~ converges to p* in Q*, then it is not difficult to show that the integrand of M e (denote it by #(x)) approximates the integrand of the error (denote it by e(x)) in the following sense. Let e > 0 be an arbitrary small number and
It is proved that [w[ tends to zero as k tends to infinity (see [182]). This fact justifies local properties of the majorant that can also be used as an efficient indicator of local errors. 6.12
Notes
for
the
Chapter
The history of the variational approach to deriving a posteriori error estimates dates back to the works of W. Prager and J. L. Synge [163] and S. Mikhlin [143]. In [143], it was shown that if a problem is stated as a minimization problem for a quadratic functional J, then the difference between J(v) and the exact lower bound provides an upper bound of the error. Since the latter is bounded from below by the values of the so-called dual (or complementary) functional, it is natural to use the respective dual problem for the computation of a posteriori error estimates (see, e.g., H. Gaevski~, n. K. Grhger and K. Zacharias [82], D. W. Kelly [118] M. Mosolov and V. Myasnikov [146]). Such bounds can also be computed by the orthogonal projection method (see S. Zaremba [221], H. Weil [218], and M. I. Vishik [215]),
208
C H A P T E R 6. L I N E A R ELLIPTIC P R O B L E M S
which creates a sequence of trial functions for the dual problem. However, in the majority of cases admissible functions of the dual problem belong to a linear manifold defined by some differential relations (we denote it Q~). In general, the determination of such functions poses a difficult task. Probably, this fact hindered the development of a posteriori error estimation methods based upon duality theory. For a class of variational problems with convex integrands dependent on the gradient of the minimizing function, a way of overcoming this difficulty was found and justified in S. Repin [168, 170, 171]. In S. Repin [174] and some other papers, such a justification is presented for a wider class of uniformly convex functionals. Functional type a posteriori error estimates for boundary-value problems with the biharmonic operator were derived in P. Neittaanm/iki and S. Repin [149]. Two-sided a posteriori estimates of the functional type were derived in S. Repin [175] for linear and some nonlinear elliptic problems. The variational approach described in Chapter 6 is based on these results. Also, [175] presents another (nonvariational) method of deriving a posteriori estimates. With the help of this method, estimates of the deviation from the exact solution were obtained for a problems that have no variational statement (see [176]). For linear elliptic equations of the second order, a numerical verification of a posteriori error estimates of the functional type was carried out in the papers by S. Repin [173], M. Frolov, P. Neittaanm~iki and S. Repin [77, 78, 181]. For equations of the fourth order, several tests were performed in M. Frolov, P. Neittaanm~iki and S. Repin [78]. For linear elasticity problems, such tests can be found in A. Muzalevskii and S. Repin [147]. Finally, we mention several results related to the subject considered, which has not been exposed in Chapter 6. Functional type a posteriori error estimates for approximations that do not satisfy Dirichl6t boundary conditions precisely were obtained in S. Repin, S. Sauter and A. Smolianski [182, 183]. Such estimates can be useful if it is difficult to precisely approximate the boundary or if a numerical method takes into account boundary conditions in a generalized sense (see, e.g., R. Glowinski, T.-W. Pan and J. Periaux [92]). In S. Repin [l $0] and S. Repin, S. Sauter and A. Smolianski [184], functional type a posteriori error estimates were used for estimation of dimensional reduction errors arising when a 3D model is approximated by a simplified 2D model.
T w o - s i d e d a posteriori e s t i m a t e s for linear elliptic problems 6.1
B a s i c relations
In this chapter, we consider problems that can be represented in the following general form: to find u E V0 + u0 such that (AAu, Aw) + (l, w) - 0
Vw E V0.
(6.1.1)
Here 1Io is a subspace of a reflexive Banach space V, A is a linear bounded operator acting from V to a HAlbert space U with scalar product (.,.), l E Vo*, and ~4 E ~(U, U) is a self-adjoint operator. We assume that V is compactly embedded in U and the operators A and .4 satisfy the relations
clliyll 2 < (Ay, y) <_ c2]Iy]l2,
Vy E H,
(6.1.2)
and ]]Awl] >_ caliwily,
Vw E Vo,
(6.1.3)
with positive constants cl, c2, and ca. We take the functional I in the form (l, w) - (f, w) + (g, Aw), where f and g are some elements of U. Such a functional is well defined and finite on the elements of V. For our analysis, it is convenient to introduce two more spaces. The quantity (.Ay, y) 1/2 determines a new norm Ill y Ill, which is equivalent to the original norm [lYll - (y,y)l/2. Another equivalent norm is lU Y Ill.= 125
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
126
(j4-1y, y)1/2, where A -1 is the operator inverse to ~4. The spaces Y and Y* contain elements of U equipped with the norms ]l]' ]]1and HI" ]]]., respectively. It is easy to see that (6.1.1) is the Euler relation for the following problem.
Problem 7).
Find u E Vo + uo such that
J(u) =
inf J(u) :--- inf P, vE Vo+uo
where 1
J(v) - -~ ]1]Av ]]]2 +(l, v). On the set (Vo + uo) • Y*, we define the Lagrangian 1
L(v, y*) - ( y * , A v ) - ~ ill Y* III2 +(l, v) and the functional
I*(y*)- inf L(v y * ) vEK
~
I
1 y, 2 ( y * , h u o ) - ~ III III. +(/,uo), l-OO~
y*EQ~, y*r
where
Q~ "- {y* E Y*I(y*,Aw) + (I,w) =O,
VwEVo}.
The problem dual to :P is as follows.
Problem 7)*.
Find p* E Q~ such that I* (p*) -
sup I* (y*) "= sup P*.
The functionals J and ( - I * ) are evidently convex and coercive on V and Y*, respectively. The sets Vo § uo and Q~ are closed affine manifolds. By Theorem 5.4.1, we conclude that there exist u E Vo § uo and p* E Q~ such that J(u) - inf 7~, I* (p*) - sup :P*. (6.1.4) The minimizer u obeys (6.1.1) and the maximizer p* satisfies the relation (Auo - A-~p *, y*) - 0,
Vy* E Q~,
(6.1.5)
6.1. BASIC RELATIONS
127
where Q~) "= {y* E Y*I(y*,Aw)= 0,
Vw E Vo}.
Note that .4An E Q~, so that, setting u0 - u, we obtain 1
I*(Ahu) = (AAu, A u ) - -~ IlI AAu Ill2,+(l, u) _ sup p*.
(6.1.6)
Since ]]I AAu III2-1R h u 1]12,we see that I*(Ahu) = J(u) = inf :P
(6.1.7)
and (cf. (5.4.11)) sup :P* - inf P. The latter relation means that 1 p, 2 1 (p*, A n ) - ~ Ill lU, +(l, u ) = ~ ]lI Au [[I2 -~-(l, U),
which is equivalent to the relation 1 n ( A u , p*) : ~ Ill Au III2
1
III
p,
2 III, -(p*,Au) = o,
(6.1.8)
or simply
p* = MAn
a.ein ~.
(6.1.9)
This is the duality relation for the pair (u, p*). Let v E 110 + u0 and y* E Y* be some approximations of u and p*, respectively. Our goal is to obtain two-sided estimates of the quantities IlI A(v - u) IiI and I]I y* - p* III that are norms of deviations from the exact solutions u and p*. We will call them deviation estimates. Deviation estimates are derived without referring to a particular numerical method used for finding approximate solutions. For this reason, they lead to a posteriori estimates valid for conforming approximations of all types. Below, we show how such estimates are derived by methods of the calculus of variations. First, we establish the following basic result. P r o p o s i t i o n 6.1.1. For any v E Vo + uo and q* E Q~, hold the following
relations: IiI A ( v - u)I~ 2 + Ill q * - P * Ill2, = 2 ( J ( v ) - I'(q')),
(6.1.10)
]lIA(v - u) ]][2 + [][ q, _ p, 1[12,= 2D (Av, q*).
(6.1.11)
128
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
Proof. In view of (6.1.1) and (6.1.5), we have
1
III A(v - u) tO2[ - J(v) - J(u) + ( A A u , A(v - u ) ) + + (l, v - u) - J(v) - J(u),
and 1
iii
q.
-
p.
Ilia, -=
I*(p*) - I*(q*) + (Auo - . A - l p * , p * - q * ) I*(p*) - I*(q*).
Since J(u) = I*(p*), these relations imply (6.1.10). For q* E Q~ the difference J(v) - I*(q*) is equal to D ( A v , q*), so that (6.1.11) follows from (6.1.10). l--I The estimates (6.1.10) and (6.1.11) are valid only for q* E Q~*, which poses some difficulties. Below it is shown how we can overcome this drawback. First, we establish one subsidiary result. Proposition
6.1.2. Let q* E Q~, v E Vo + uo, ~ E R+, and y* E Y*. Then
J(v) - I* (q*) ~_ _< (1 + ~)D(Av, y*) +
(
1+
~ 111
-
tit..
,
)+(iv ,
(6.1.12)
Proof. For any y* E Y*, we have J (v ) - I" ( q" ) - 1(611Av I[Lu +
I]1Y"
]612)+
] q, ili~,-Ill y, +~(Ul
n l2, ) - ( A ~ o q *
-
u0)
Since (l, v - uo) - (q*, A(u0 - v)), we find that J(v) - I* (q* )
1
- ~ (Ui h~ iii: + IlLy* lil~,) - (q*, A~) + 1
+~ (Ill
q,
2
ili, - nl
y,
ni~,) -
1 q, y, 2 = D ( h v , y*) + (y* - q*,Av - A - l y *) + -~ [[[ [1[,"
6.1. BASIC RELATIONS
129
This relation yields (6.1.12) if we note that 2 (y* - q*, Av - A -~ y*) _/3 ]1]hv - A -~ y* [1[2 +/3 -~ Ill Y* - q* [][2,. E] Introduce the quantity
d~(y*)'- ~,~Q; inf Ill q* - y * 1112,9 Then, (6.1.10) and (6.1.12) imply the estimate
1
2 111A(v- u)Ill2<_ (1 + 13)D(Av, y*)+
+
(6.1.13)
2d~(y *)
with v E V0 + u0 and y* E Y*. From (6.1.13), it follows that 1
III A ( v - ,.,)1112< M(v, fi),
V,, e Vo + uo,
,8 e ~ ,
where j~4(v,/3) "- y'ev*inf { ( 1 + fl)D(Av, y * ) + (1+ ~ ) l d ~ ( y * ) } . Let us show that this relation holds as equality. Proposition
6.1.3. For any t3 E I~, 1
III A(v- u)iIi ~- .M(v, Z).
(6.1.14)
Proof. Set y * - ~ p * + ( 1 - ~)AAv. Then D(Av, y*) = 2 A2 INh ( v - u)][]2. Since d~ (y*) _<][] P* - Y* ]U2,= = (1 - A) 2 U[P* - Ahv [[12,- (1 - A) 2 Ill Ah(u - v)[[I,2= = (1 - A) 2 [1[A ( u - v)][i 2, we obtain
~4(~, ~)<
1 ((1
+/x
i1
A/v
/2
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
130
The right-hand side attains its minimal value at A - 1/(1 +/3), which leads to the estimate 1 III i ( v - u)11[2>_ M(v,f~), Vv E V0 + uo, /3 E 11~. Recalling that the inverse inequality has already been established, we arrive at (6.1.14). V1
Proposition 6.1.4.
{1
sup
d~(y*)-
- ~ lUAw
}
lU= -(1,w>- (y*,Aw) .
(6.1.15)
wE Vo
Proof. This assertion is a consequence of the basic duality relation inf P sup P*. Indeed, -
III
-
}
1112, - -
up ,'eQt-v"
-
1 ~?, 2} III III 2 * '
where
Q r - v, := {,* e Y*l,* = ~ , _
y,,
~, e Q;}.
In other words, ~* E Q~ - y * if (7", i w ) = -(l, w) - (y*, Aw),
Vw E Vo.
The right-hand side of this relation is a linear continuous functional. We denote it by 1v and rewrite the relation as follows: (~*, Aw) + (lu, w) = 0 Vw E Vo. Then, Q~ - y* - Q~* and ida(y*)--sup { 1 r]* } . ,, eQr~ - ~ lU 1112, This maximization problem coincides with Problem P* if we set Uo = 0 and l = lv. From this fact, we conclude that inf~evo ~1 Ill Aw I[12+(l v, w) } -
ld~(y*)--
1
= - wevoinf ~ Ill Aw [1[2 +(l, w ) + (y*,Aw) =
sup
{1 -~
Ul Aw III2 - ( z , w) - ( y * ,
.
wE Vo
[:]
131
6.2. TWO-SIDED E S T I M A T E S
Corollary 6.1.1. Propositions 6.1.3 and 6.1.4 lead to the conclusion that the functional A4(v, f~) has a minimax form
M(,,Z) =
( ( 1)( )} inf sup (l+~)D(Av, y*)+ 1 + ~ -(y*,Aw)-g(w) . yEY* wEVo (6.1.16) This relation is crucial for deriving upper and lower bounds of deviations.
6.2
Two-sided
6.2.1
estimates
of deviations
Upper estimates
In view of (6.1.16), we have j~4(v, f~)<_ (1 + ~)D(Av, y*)+ +
(
i)sup{ 1 * 1 + ~ weVo - 2 Hlhw1112 -
},
where y* E Y*. Let A* denote the operator conjugate to A, which is defined by the relation (y*, Aw) = (A*y*, w),
Vw e Yo.
Then
IllllVo*llwilv + Ily*llllAwl[ ~_ (C311111iv(~ Jr-liy*[I)liAwll _~ < ~-~/2 (~xlllllvo. + Ily*ll) INA~ I11.
(6.2.1)
132
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
Hence, the value of the supremum in (6.2.1) is finite and sup
(1 -
IIIAw I[I~ - ( Z , w ) -
(y* Aw)
}
(
I 1 } sup - [1[Aw III2 +IZ+A*y*IIIAwll <
< --
w E Vo
-2
w E Vo
-2
sup ~>0
'
+1/+
2
y'It
-
2
-
-
(6.2.2)
If+A'y*[ 2
Define the functional 1 + h* y ,j2 . 1 + ~1 ) -~[l
u~vt@(v,/3,y*) "- (1 +/3)D(Av, y*)+ By (6.1.16) we see that 1
[][ A ( v - u)]]]2_ .M(v,~) < M~(v,13, y*).
(6.2.3)
Thus, for any y* E Y* and/3 > 0. the functional M~(v,~,y*) is a majorant that provides an upper bound for the deviation v - u evaluated in the natural energy norm. 6.2.2
Lower estimates
Since .M(v, fi) _ sup
inf ((1 4- /3)V(Av, y * ) -
w E V o y . EY*
-
(
1)(1
1+ ~
~ Ill Aw Ill2 +q, w) + (y*, Aw)
)}
,
we have M(~,fl) >
inf
y*EY*
{
(1 + fi)
(1)(~)
~ IIIy* [112,-(Y*, hv)
+(1+/3)~11]Av1112-
-
1+
( 1)(1 1+~
(y*, hw)
}
+
~[l[Awlll 2+(l,w)
)
,
6.2. TWO-SIDED E S T I M A T E S
133
where w is an arbitrary element of Vo. Evidently, this estimate is also valid for f~w, which yields
:nf.{' ~111
"
III.-(y,h(v+~,))
A/I (v, f~) >_ (1 + f~) ,,eY
}+
+(1+,8) (~ UlAv III~ -~ III Aw I!!2 -(l, ~)). Note that inf
y*ffY*
1 y, , } _ Ul Ill~,-(y A(v+w)) > _> y.inf ~y. {1~ III y , 1112.- III y, III. ill A(v + w) III - - 5
1
Ill A(v + w)
Ill~
Thus, we obtain
M(v,~) >
1
(1 +/~) -~. III A(v + w)III ~ + 1
/3
2
+ ~ III A~, III2 - ~. III Aw III - (l, ,.,.,)} = (1 + Z){-(.AAv
A~)
1 + Z III Aw III~ -(l ~)} 2
~
"
Here, w is an arbitrary function in Vo. Therefore, we may replace w by w/(1 + fl). Such a replacement leads to the estimate
89III A(v- ,.v)II1~> Me(,,, ~),
w e Vo,
(6.2.4)
where the minorant is defined by the relation 1
Me(v,,.v) := - ~ III h,.,, III2 - ( A A v , Aw) - (1, w). Remark 6.2.1. Assume that v coincides with u. In this case, 1
1
M e ( U , w) - - ~ Ill Aw lI]~ - ( A A u , A w ) - (l, w) - - ~ II hw ILl2 and, therefore, sup A4e(u, w) = O. wE Vo
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
134
The same is true for the majorant. Indeed, set ~* - .AAu. Then, ~4,(u,~,~)-
(1 + f l ) D ( A u , ~ ) +
( 1)1 1+ ~
~]l+
,
where the first term is zero and
(l, w) + (AAu, Aw)
IZ + A*~*I = ~up so that it is zero as well. Thus,
inf .M~(u, ~, y*) = O.
y*EY*
6.2.3
Estimates of deviations in terms of the dual variable
Let y* E Y* be an approximation of p*. For any q* E Q}', we have
lily -p'Ill2, <(l+'y) llly -q'1112,+ 1 + ~
W -
Ill,,
(6.2.5)
where ~' is a positive number. From (6.2.5), (6.1.10), and (6.1.12), we obtain
IIIy* -p* III,~ < (1 + ~)III y* -q* 1112,+ +2
(11+ ~)
=(1+7)
(
(J(v)-
1 1+-+
1)
,,(q*))= y,_q, I1[ {]1, 2+
+2(1+~) ( 1 + ~1 ) D(Av ,y *) . Therefore, 1-III y*-p* 1112, <- ( 1 + 7 ) 2
1 1 + - ~+
1)1 3~
~ d~(y*)+
+(1+ f ~ )(1 + ~ 1) D(Av'y*)
(6.2.6)
and ~-III y*-p* III,~ < ( 1 + ~ ) 2 +(1 + ~)
1 1 h,y,12 + 1+-+~ ~1 ) ~lz+
(11+ ~)
D(hv, y*).
(6.2.7)
6.3. P R O P E R T I E S OF TWO-SIDED E S T I M A T E S
135
Rewrite this estimate as follows:
89IIIy* - p* 1112,_~ . ~ (y*, ~, ~, ~),
(6.2.8)
where f14~ denotes the right-hand side of (6.2.7). This estimate holds for any y* E Y*, positive parameters/3, 7, and any v E V0 + u0. It is important to outline a certain duality of the estimates (6.2.2) and (6.2.8). In (6.2.2), the majorant contains a "free" function y* that belongs to the space Y* of the dual problem. The minimization of A4r over Y* gives the sharpest upper bound of the deviation evaluated in the energy norm of the primal problem. In (6.2.7), the majorant J~4~ contains a "free" function v, which belongs to the set Vo + u0 of the primal problem. Minimization over this set leads to the sharpest upper bound of the deviation evaluated in the energy norm of the dual problem. Estimates of the deviations in the dual energy norm may be useful in problems arising in continuum mechanics, where the dual variable often has an important physical meaning (e.g., in solid mechanics it is a stress function).
6.3 6.3.1
Properties
of two-sided
estimates
Exactness
Values of ~4o(v, w) depend on the choice of w and values of A/Is(v,/3, y*) depend on f~ and y*. Let us show that w, f~, and y* can always be chosen such that the minorant coincides with the majorant and, consequently, coincides with the energy norm of (v - u). Since [l + A'p*[ - sup (l, w) + (AAu, Aw) _ O, ,~
Vo
IIIhw III
we find that .h4r
- (1 + ~)D(Av, p*) = (1 + f~)
(1 1 ~ lUAv Ill: + ~ ]ll Au Ill2 - ( A A u , Av)
m
1+/~ ][] A(v - u)]]12 2
136
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
and, therefore, 1 ~ Ill A ( v - u)III 2
inf M e ( v , ~ , y * ) y*EY* f~ER+ Similarly,
~) -
Me(~,~-
1
- -~
2 III A(u - v) Ill= - ( .aAv, A(u - v) ) - (l, u - v) 1
-
1
I!1A(u - v) III2 + ( A A ( u - v), A(u - v)) - ~ III A(u - v) Itl2
Thus, the estimate (6.2.4) is exact the in sense that
~sup Vo Me(v,w)-
~1 Ill h ( u -
v)III 2
Let (Yi*}~l and {Voi}~l be two sets of finite-dimensional subspaces. We assume that they are dense in Y* and Vo, respectively. The latter means that, given c > 0 and arbitrary elements y* E Y* and w E V0, one can find a natural number k~ such that inf I1~ - wllv < ~, ~eVoi -
inf III ~* - y* II1< 6, ~*eL* -
Vi > k~. -
(6.3.1)
The methods of creating sequences of finite-dimensional subspaces with above-mentioned property are well known in numerical analysis. In particular, they are represented by regular finite element approximations. Let us show that two-sided bounds can be evaluated precisely by minimizing the m a j o r a n t on {Yi*} and maximizing the minorant on {Voi}. Take a small e > 0,. Then there exists a number k and elements Wk E Vok and p~ E Yo*k satisfying the conditions
Ilwk - ( u - v)llv ___~,
lUp~ - p* Ill, ~ ~.
(6.3.2)
Define two quantities M~-
inf
.Mr
(6.3.3)
f~ER+ and
M~-
sup M e ( ~ , ~ ) . w~ EVok
(6.3.4)
6.3.
PROPERTIES
OF TWO-SIDED
137
ESTIMATES
From (6.3.3) and (6.3.4), it follows t h a t
M ~ _< M ~ ( ~ , ~ , p ~ ) ,
M ~ _> M e ( ~ , ~ ) .
(6.3.5)
1
Let us show t h a t f14~ and f14~ tend to ~[[A(v - u)[[ 2 as k -+ oo. First, we the upper estimate first. We have M~(v,~,pk)
,
-
, + ( 1 + ~1)1L-~lt + A * Pk"12 9 (1 + ~ ) D ( A v , p~)
(6.3.6)
Here
It + A*p~ I - sup (p* - p~' A~o) < ~ eVo IIIA~o III -
(6.3.7)
and I(A v - ..4 -1 Pk,9 A A v D(Av, P k9) - -~
- P k9) --
= 1 (A(v - u ) - A - ~ (p~ - p * ) A A ( v - u ) - (P*k - P * ) ) 2 1 9 p, 2 9 p, -- ~ Ill h ( v - u)Ul 2 + Ill P k Ill, - ( h ( v - u ) , P k -- ). From the latter estimate we see t h a t D(Av,
1 p ~ ) _ ~ Ill A(v - u)
Ill2 + c
1 2 Ill A(v - ~) Ill + ~ 9
(6.3.8)
Combining (6.3.6), (6.3.7), and (6.3.8), we obtain
M~ < Me(~,~,p~)= = (1 +
(1 1~) e) ~ IIIA(v- u)III2 +6 IIIA(v- u)III + ~ + 1 1 + ~6(1 + e) - ~ Ill A(v - u)III 2 -~-C4C-[- O(~),
(6.3.9)
where c4 - ~x (1+ III A(v - u) III)2 . Recall that for any small e there exists a n u m b e r k such t h a t the estimates (6.3.2) hold. Thus, we conclude that
M~
1
~ ~ IIIA ( v -
u)III 2
as
k ~ c~.
(6.3.10)
Remark 6.3.1. Note t h a t the constant ca depends on the norm of ( v - u), so t h a t for a good approximation this convergence is faster t h a n for a bad one.
138
C H A P T E R 6. L I N E A R ELLIPTIC P R O B L E M S Now, let us focus our attention on lower estimates. It is easy to see that 1
M e (~, w~) - - ~ Ul Aw~ Ill2 - (AAv, Awk) -
1
1
_> ~ Ill A ( u - v)Ill2 - ~ c 2 | A ( w k -
( u - v))Ill ~
This implies the estimate 1 Ill A ( ~ -
~)Ill2_> M ~ >_ ~1 IIIA(u- v)lU2 -c5 c2,
(6.3.11)
where c5 is a positive constant dependent on the norm of A. The estimate (6.3.11) shows that
1
M ~ ~ ~ Ill A(u-v)LII
2
as k --+ oo.
(6.3.12)
Remark 6.3.2. Having A/[~ and f14~, one can define the number ~} : - f14~ -> I,
(6.3.13)
which gives an idea of the quality of the error estimation. From (6.3.10) and (6.3.12) it follows that ~k -+ 1,
6.3.2
as k --+ +oo.
Computability
By computability we mean that upper and lower estimates can be computed with any a priori given accuracy. The quantities ~4~ are computed by solving finite-dimensional problems for a quadratic integral functional. Such a task is solvable by well-known numerical methods. However, the quantities A4~ are related to the majorant A4r whose second term is represented by the negative norm. In general, computing such a norm may be very expensive. Below we show that under certain assumptions (which hold in the majority of practically interesting cases), this term is estimated by an explicitly computable quantity. This gives a directly computable majorant that preserves all of the main properties of A4r
139
6.3. PROPERTIES OF TWO-SIDED ESTIMATES Assume that l E U
(6.3.14)
and the variable y* belongs to the set
Q*:={y* eY*l
A'y* EU}
lCU,
and
so that for any w E V0, we have (l + A'y*, w) = (l + A'y*, w). In this case, (l + A'y* w)
II + A*y*I = sup ~,~vo
' III Aw III
< sup - ,o~vo
Ii/+ A*y*II Ilwll III Aw III ii
< II/+ A*y*IIc~ -~
< -
Ii
sup ~
wE Vo iilXWll
< c11c~1111 + A*y*II.
Here cl and c3 are the constants in (6.1.2) and (6.1.3). Now, the majorant is represented in the form
Mr
=
1 ) c2
= (1 + f~)n(Av, y * ) +
1+ ~
71It + A'y* II2
(6.3.15)
where c2 - c-~2c32. If l E U, then p* E Q* and, therefore, inf
y*EQ* ~>0
M~(v,~,y*) < Mr
(1 + r
1
Iii A(u-v)III
2,
where c > 0 may be taken arbitrarily small. Hence, we arrive at the following result. P r o p o s i t i o n 6.3.1. For any v E Vo + uo, ~ E I ~ , and y* E Q*, the following relations holdi
1
III A(u-v)lU=_< M r 1
Ill A ( u - v ) U l
2-
< ir
(6.3.16)
inf Mr
(6.3.17)
y*EQ* f~>0
These relations mean that the majorant M s is also reliable and exact. However, in contrast to j~lr it contains the norm of a Hilbert space U instead of the norm of Vo*. In particular problems, the former is usually
C H A P T E R 6. L I N E A R ELLIPTIC P R O B L E M S
140
given by a directly computable integral, while the latter one is the norm of a Sobolev space with negative index. Obviously, this makes M e much more attractive from the viewpoint of practical applications (see Section 10). To justify this conclusion, we prove that a convergent sequence of upper bounds can be constructed by a sequence of finite-dimensional problems associated with the majorant M e. Assume that {Q~ }~=1 c Q* is a sequence of finite-dimensional spaces, which are dense in Q* in the following sense: for any e > 0 and any q* E Q* one can find a natural number ke such that inf [Iq~n- q*[[Q* ---e, qTeQ~n
Vm > ke,
(6.3.18)
where [[q*[[Q. := [[q*[J + [[i*q* [[. P r o p o s i t i o n 6.3.2. If the spaces ln*xoo t'~ kJk=l are dense in Q 9, then inf M e(v,/3, Ym) - 21 [[[ A ( v - u)[1[2. mlim - ~ u;,eQL
(6.3.10)
r
Proof. Take an arbitrary small r > 0 and find a respective positive integer ke. For m > ke, we have inf M e ( v , ~ , y m ) < M~(v,r y;,~Q;,
)=
BER+
= (1 + e ) D ( A v , p;~) +
1+ ~
7111+ A*p;~[] 2,
(6.3.20)
where p ~ e O~ is an element such that Ilp;~ -P'lie" -- 2e. Since p* e O*, the existence of p ~ follows from (6.3.1S). The first term is easily estimated"
D(Av,p~)-
1 1 ~ I[I Av ][]2 + ~ [][p ~ [][2 - ( p ~ , A v )
__
1 p, , , _ <_ D(Av, p*) + -~ (.A-l(p*~ ),Pine + P*) + (P* -P~nc Av) < 1 1 , p, , p, < ~ [Hh (v - u)) [[[2 + 2 [[I P.~ J[I. (N[ P,~ + [1[. +2 [~ hv [[[), Hence, 1
D(Av,p*~) <_ ~ ]1]A ( v - u))]]12 + # ~ . For the second term we have []/+ A*p~[[ < [[A*(pm~ -P*)[J-< [ [ P ~ - P*J[Q* < #2~,
6.4. ELLIPTIC EQ UATIONS OF THE SECOND ORDER
141
where #1 and #2 are positive constants. Therefore, 1
M e ( v , E , p ~ ) <_ (1 + E)~ Ill A ( v - u)Ill 2 + I
)(e + e') -
IIIA(v u)III2 +o(e'). (6.3.21)
Now, (6.3.20) and (6.3.21)imply (6.3.19).
[:]
+ (~ +
-
In concrete problems, the majorant M e is a quadratic integral functional. Therefore, the numbers M~-
Me(v,/3, y;)
inf
(6.3.22)
/SeR+
can be found by direct minimization procedures. They form a computable sequence of upper bounds such that M~
1 > ~ Ill A ( v - u)Ill 2
as
k --+ c~.
(6.3.23)
Recalling (6.3.12), we conclude that 1
M~ ~ ~ Ill A ( v - u)1l[2<_ M~.
(6.3.24)
The quality of these bounds is estimated by the quantities
~k
"'--
(6.3.25)
z4~'
which are computable and tends to 1 as k -+ +c~.
We can call them
computable effectivity indexes.
6.4
Linear
elliptic
equations
of the second
or-
der Let f~ be a bounded connected domain in IRn with Lipschitz continuous boundary 0f~, V = H1 (f~), U = L2(f~, IEn), and hw be the gradient operator Vw-
(o,l,oxl, ' " , oxn~ . Define a symmetric matrix A aij E L ~
{aij }such that (6.4.1)
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
142 and
cx]~] 2 < aij~i~y _ c2]~i 2,
V~ e lI~~.
(6.4.2)
Then, the spaces Y and Y* are equipped with norms
Ill Y I I I 2 - / A y ' y d x ,
lilY* 1112-/A-ly*'Y *dx.
~2
6.4.1
~2
Dirichlet boundary conditions o
Let V0 - H1(~2). The relation
f AVu. Vw dx + (l,w)
O, Vw E V0,
(6.4.3)
~2
defines a solution of the boundary-value problem div
AVu- f
(6.4.4) (6.4.5)
in ~,
u - uo
on 0~.
The relation (y*, Aw) - ( A ' y * , w) comes in the form
y* . Vwdx = ( - d i v y * , w ) ,
(6.4.6)
fl
where A* - - div and the function div y* is viewed as an element of H -1 (~). The inequality (6.1.3) is the Friederichs inequality O
c~llVwl] >__llw]l,
Vw e Hi(D).
(6.4.7)
Upper estimates of NIv - u III for an approximation v E Vo + uo follow from (6.2.3). We have 1/AV(v-u). 2
V(v-u)dx
< j~4~(v, f~, y*) '
where
Mr
= 1+ Z 2
f n
(Vv -
A - l y * ). ( .AVv - y* ) dx +
1~~ ] div y,_ f ,~
6.4. ELLIPTIC EQUATIONS OF THE SECOND ORDER
143
and
f y* . Vwdx + (f,w} [divy*-f[-
sup (div y* - f, w) = sup fl ~ Vo
Ill A ~ III
~ e Vo
Ill A ~ Ul
Assume that f E L2(f~) and y* E H(f~, div), so that
(f,w)-/fwdx,
/y'.Vwdx--/divy'wdx.
f~
f~
f~
Then, the negative norm has an upper estimate Cn
[divy* - f[ _< ~-1[ ]div
y,
- f]l.
Hence, we arrive at a particular form of the majorant (6.3.15), which yields an estimate
l l AV(v_u) V ( v - u ) d x < 2 tll
f~
l+Of
-
( V v - A-ly*) 9(AVv-y*)dx+
2
f~
1 +Z c~ [[ div y* - f [[~ 2/3 cl
(6.4.8)
valid for any v E Vo + Uo,/3 E I ~ , and y* E H(D, div). Let {Yk*} be finite-dimensional subspaces of Y* such that Yk* E H(12, div) forall k = 1,2,...; dim Yk* -4 +oo as k -+ oo. By (6.4.8) we obtain computable upper bounds
inf {1+/~/
.M~ m
Y~eYk* BER+
2
( V v - A-ly*) . (AVv-y*) dx+
fl
1+~ c~ I[ div y * - f [l~ ~.
2/3 c~
(6.4.9)
J
Lower estimates follow from (6.2.4). We have
12/ AV(v - u) . V(v - u)dx _>Me(v, w), f~
VwE Vo,
(6.4.10)
C H A P T E R 6. L I N E A R ELLIPTIC P R O B L E M S
144 where
1/
M e ( v , w) - --~
A V w . Vw dx -
/
A V v . Vwdx - (f , w).
gt
Let {V Ok} be finite-dimensional subspaces such that
V~k E Vo forall k - 1,2,...; dim Vok -+ +c~
as k -+ c~.
Solving finite-dimensional problems, we find numbers
M~ -
sup M e ( v , wk).
(6.4.11)
w~ E Vok
1 Both sequences M~ and MS tend to ~ Ul~ -
~ Ul~ ~s k -~ ~ , provided that
{Yk*} and {Vok} possess necessary approximation properties (see Section 6.3). Note that if v is a Galerkin approximation computed on Vok, then M e ( v , Wk) -- O. This means that to obtain a sensible lower estimate in this case, one must always use a finite-dimensional subspace that is larger than Vok.
6.4.2
Neumann boundary c o n d i t i o n
If (6.4.5) is replaced by the Neumann boundary condition
v. A V u + F -
0
on
0~t,
(6.4.12)
where v is the vector of unit outward normal to 0~, then the general scheme can be applied if we set
vdx - 01 and define A'y* E Vo* by the relation (A* y*, w) - / y* . V w dx,
Vw e Vo.
Assume that f E L2(D),
F E L2(0gt)
(6.4.13)
6.4. ELLIPTIC EQUATIONS OF THE SECOND ORDER
145
and that the equilibrium condition
I
(6.4.14)
f dx + / F dx - 0
gt
Oft
holds. Then the problem (6.4.4), (6.4.12) has a solution defined by the integral identity
/ A V u . Vw dx + / fw dx + / Fw ds - 0, Ft
~2
Vw E Vo.
(6.4.15)
Oa
In the case under consideration,
(l, w) - / f w dx + / Fw ds a
of~
and
(l +
- /(y*.W
+
~2
O~
In general, ]l + A'y* I is estimated in terms of the norms !] divy* - f IIH-1 and
II y*.u + F IIH-1/2.
However, if we assume that y* possesses a certain regularity, so that y* e Q * ( g t ) ' - {y* e Y* I divy* e L2(~),y*.v e L2(0gt)}, then (l + A'y*, w ) = / ( f - d i v y * ) w d x f~
+ / ( F +y*'u)wds oa
and, therefore, I(l + A'y*, w)l _< _ I] div y* - f [12,a]lwl]2,a + Ilu y* + FI]2,oa]lwll2,oa. (6.4.16) Let the constant ca be defined as
f AVw. Vw dx
i
-
inf
a
2
(6.4.17)
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
146
Since the trace operator is bounded, this constant is finite. (6.4.16) implies the estimate
Therefore,
Ill + A'y*, w)J < <- ca ([[ div y* - /II 22,~ + II~ y*
+ FII2,0n) ~/2
Ul Aw Ill2
and the second term of the majorant is calculated as follows" [l + A*y*J = sup
wEVo
(l + A*y*,w) < I][ h w It[
-
<_ c~ (11 div y* - f I12 2,~
+ I1~'y* +FIJ~,0n) 1/2
The term D(Av, y*) is defined in the same way as in the Dirichlet problem. We see that the distinction in the structure of M e for the two main boundary-value problems manifests itself in different values of cn and in the fact that the Neumann problem majorant contains an extra term IIv y* + FJl2,0n that penalizes violations of the boundary condition (6.4.12).
6.4.3
Mixed
boundary
conditions
Let 012 consist of two measurable nonintersecting parts 0tQ and 02Q, on which different boundary conditions are given:
u-uo
on
0112,
v. A V u + F = 0 on
(6.4.18) 02~.
(6.4.19)
Set V0 : : {v ~ H ~ ( n ) l v - 0
on
0112}
and (h* y *, w) = / y*. Vw dx,
Vw e y0.
Assume that f e L2(12),
F e L2(0212).
Then, the functional I has an integral form
(l,w) = / fwdx + / Fwds. gl 02~2
(6.4.20)
6.4. ELLIPTIC EQUATIONS OF THE SECOND ORDER
147
For any w C Vo, we have (I + A*y*,w>-
f (y* . Vw + fw)dx + / Fwds = o,n
=/(divy*+f)wdx+ /(y*.v+F)wds, ~2
02 ~2
provided that y* possesses an extra regularity, namely, y* e Q*(i2):= {y* e Y" I divy* e L2(12),
y*.v e L2(0212)}.
Note that in view of (6.4.20) the exact solution p* belongs to the set Q*(12), so that the above-made assumption on the admissible set of y* is not restrictive. Now, we obtain
I
-
.fll~,~ll~ll~,n + lly* 9 v + Fll~,o~ll~ll~,o,n.
Let 7 and 7, be two numbers such that 7>I,
7,>I,
1
1
--F--=I. 9' 7,
Then
I
fll~,~+'Y, Ily* u+Fll~,a~) ~/2
x ~llwll~,a+--Ilwll~,o,a~, Since
I1~11~,~ < c[(~)llVwll ~ --
2,fl~
Vw e go
(6.4.21)
and
2 <_ c2(~,o2~)llwll~,~,n , I1~11~,o~,
w
e Vo ,
(6.4.22)
we find that 1
I7 llwll~,n + --llwll~,a,~ _~ ~',
< ( c~(a)~1 + c~(a, o~a) (~ + c[(a)) -:1 ) ,,Vwi, 2. ,
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
148
Therefore, there exist a positive constant C3 (which depends on ft, 02fl, and 7) such that
f A V w . Vw dx a
1 =
inf
1
2
i
2,ft
,y,
Moreover,
( C ~ ( a ) ~1 + C~(ft, O2ft)(1 + C~(a)) 7--1) c~-~'
C3 _
(6.4.23)
so that the constant C3 has a computable upper bound, provided that the constant C1 (it) in the Friederichs inequality (6.4.21) and the constant C2(fl, 02fl) in the trace inequality (6.4.22) are known. It is worthyof note that these constants depend only on ft and 02ft. Now, we find that I(/+A*Y* , w ) l <_C z ( 7 1 1 d i v y * - fl[ 22,a + 7, IJy*'u + F II~,o,a) 1/2 lU Vw Ill. From this estimate, we obtain I/+A*y
_< 2 f ~ ) <_ < c3 (711divy* - f]l~,a + 7,11Y*" ~' + FII2,o,
<_ c~(a)~ + c~(a,o~a)(1 + c~(a))7 7 1 x
711divy* - fll~,a +
7
7
1
•
Ily*"u + FII2,o~a cF 2.
(6.4.24)
A minimum of the right-hand side of (6.4.24) is attained if 7-~'-1+
Ily*. u + Fll2,o2nCl(a)
il div y* - fil2,ac2(a, o~a)(1 + c[(a))~/~
In this case, II + A*y*l 2
< (C1 (ft)]1 div y* - fl]2,n+ \
+C2(a, 02f~)(1 + C2(fl))l/211y*'u + Fll2,0~a
cf 2. (6.4.25)
6.4. ELLIPTIC EQUATIONS OF THE SECOND ORDER
149
Now, we can present the majorant in two forms. The first one uses (6.4.25), which yields
M e ( v , ~ , y , ) _ 1+2 ~ f (Vv
-
A -1 y *) 9 (AVv
-
y* ) d x +
1 +/~ (C1 (f~)ll div y* - fll2,f~§
+ 2Z
+ c2(a, o~a)(1 + c~(a))~/~lly* ~, + FII2,o~a) 2C l 9
2.
(6.4.26)
If we take 7 - 7* - 2, then the majorant has a simpler form
Me(v ' ~ , y , ) _ l +2 /3 f (Vv
-
A -1 y *) 9 (AVv
-
y* ) d x +
fl
1+ (C~(a) + C~(f~ 02a)(1 + C~(a))) x + 2~ ~ ~ 2
•
2 ) 2,a + IIy, "u + FII2,o2a
9
(6.4.27)
This majorant gives an upper bound of the deviation for any v C V0 + u0, y* C Q*, and ~ > 0. It is easy to see that it vanishes if and only if v = u and y* - AVu. Also, it is obvious that the majorant is continuous with respect to the convergence of v in V and y* in Q"
Remark 6.4.1. The estimate (6.4.26) remains valid for the Neumann problem. 6.4.4
Lower
estimates
Lower estimates for the problems considered follow from (6.2.4). They have a common form:
1 f AV(v - u). V(v - u)dx > lk4e(v, w) 2 '
V w e V0,
f~
where
if
Me(v, w) - --~
f~
AV(w - v) . Vw d x -
f
f~
fw d x -
f
Fw ds.
09f~
Here Vo depends on the type of boundary conditions, and the integral over 02f~ must be eliminated in the case of Dirichldt problem.
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
150
6.5
The linear elasticity problem
Assume that f~ satisfies the same conditions as in the previous section and 0f~ consists of two disjoint parts 01 f~ and 02~ and [01 f~[ > 0. The classical statement of a linear elasticity problem is as follows: to find a tensor-valued function a* (stress) and a vector-valued function u (displacement) that satisfy the system of equations a* = L e
in
f~,
(6.5.1)
div a* = f in
f~,
(6.5.2)
0xfl,
(6.5.3)
u=uo
on
a*u + F = 0 on
02~2.
(6.5.4)
Here f and F are given forces and L - {Lijkm} is the tensor of elasticity constants, which is subject to the conditions Cl1r 2 _< L e ' r
_< C2[r 2,
Vr e M[~s•
(6.5.5)
and
Lijkm = Ljikm = Lk,nij,
Lijk,~ C L~(f~).
(6.5.6)
Henceforth, we assume that f e L2(f~,I~n),
F E L2(02f~,I~n).
(6.5.7)
Then, a generalized solution u E V0 + uo is defined by the identity
/ Le(u) " ~(w) dx + (l,w> - O,
Vw E Vo,
(6.5.8)
f~
where
(t,w) - / . f .wdx + / F.wds. f~
02f~
The existence and uniqueness of u follows from (6.5.5) and the Korn's inequality (see, e.g., [64]). Let v and y* be some approximations of u and a*. Estimates of v - u and y* - a* follow from the general scheme if we set U - L2(D,M~, •
V-Hl(f~,I~n),
Vo - {w E V l w - 0 on 0112},
III y Ill f Ly. f~
Uly* II1.- f f~
y*
151
6.5. THE LINEAR E L A S T I C I T Y P R O B L E M
and identify Av with the symmetric part of the gradient tensor
e(,)- ~1 (w + (w) ~) In this case, (A'y*, w) - f y* " e(w) dx,
(6.5.9)
Vw e Vo,
where two dots stand for the scalar product in ~/~nxn. If y* e Q* "- {y* e Y* I divy* e L2(~2,~•
y*v e L2(02~, I~)},
then (l+ A * y * , w ) - - / d i v y * . w d x + / ( y ' u ) . w d V . ~2
6.5.1
(6.5.10)
82i2
Upper estimates
By (6.2.3) we obtain the upper estimate of the deviation from the exact solution in terms of "strains"" 1 / L~(v - u)" 6(v - u) dx < .Ms (v, ~, y*) 2
-f~
where .Ms (v ~, y*) - 1 +/3 D(ev y*) + 2
l~Z I A'y* + l I:
and 1 /(iL~(v).e(v)L_ly,.y,_ D(e(v), y , ) - -~
2r
y*)dx -
f~
=
f(e(~l-L-~ v*).(Ls(~)-v*)a~. f~
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
152 If y* E Q*, then IA*y * + l l -
sup
(A'y* + l, w>
ill h~ ill
wE Vo
f (y* "~(w) + f . w) dx + f F . w ds :
sup
iii ~ (~) ill
wE Vo
f(f-divy*).wdx+ f - sup
f~
~Vo
(F+y*u).wds
02 ~
III~(w)III
Let Ca be a constant in the inequality
f lwl dx + f Iw[2 ds <_C~lle(w)ll~, ~2
Vw e Vo.
(6.5.11)
02f~
Note that the existence of such a constant follows from the Korn's inequality. By (6.5.11), we obtain
IA*y*+ll< < (lldivy*-fllg+llF+y*vll20,a) 1/2 sup (liwllg+llwl Io~a) 2 1/2 < ,~eVo IIIe(w)III <_ Ca~Y ~/~ (LIdivy *-/ll~ § iIF + Y*"11~2~) ~/~ and arrive at the majorant M e"
M e (e(v), y*) - 1 +2~ f ( c ( u ) - L_ly , )" ( L c ( u ) - y. ) dx+ 1 +r C~ (11 div y* -fll~+llF+y*vllo2n). 2 + 2~cl
(6.5.12)
It has a clear physical meaning. The first term of M e is nonnegative and vanishes if and only if
y* - L~(~). It penalizes violations of the constitutive relation (Hooke's law) (6.5.1). The meaning of the second term is obvious: it contains L2-norms of other two relations (6.5.2) and (6.5.4), which gives errors in the equilibrium equation
6.5. THE LINEAR E L A S T I C I T Y PROBLEM
153
and boundary condition for the stress tensor. Thus, the majorant not only gives an idea of the overall value of the error, but also shows its physically sensible parts. The latter information may be of utmost interest, because it suggests a correct way of finding a better approximation. Let {Yk*} C H 1(~, MI'~x'~ ) be a collection of finite-dimensional subspaces that satisfy the condition (6.3.18). Then, (6.5.12) provides a sequence of computable upper bounds
M ~ ~"
inf { l + / 3 f v" eY~ 2
(L~(v)'c(v)+L-ly*'y
*- 2e(v)'y*)dx
y,
,vl[2 } o n) ,
1 +/3
+ 2/3cl C~
(lldiv
-
Yll
+ IIF + y
(6.5.13)
which tends to the exact value of the error.
6.5.2
Lower
estimates
Lower estimates follow from (6.2.4). We have
1 f L~(v - u)" ~(v - u)dx > .A4e (v, w) 2 '
VwE V0,
where Me(,.,,,-,.,)
-
1 fLe(w)"e(w)dx-fLc(v)"E(w)dx-
2
~2
~2
-fy.wdx-fF.wds. 02 g~
Maximizing the functional Ado on a certain finite-dimensional space Vok C V0, we obtain a computable lower bound A4o-
sup A40(v, wk). wE Vo~
If the spaces Vok satisfy the conditions stated in Section 3, then the sequence of numbers {A4e) tends to (v - u)Ill2 (cf. (6.3.12).
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
154
6.6
Linear elliptic equations der
o f t h e f o u r t h or-
In this section, we consider the problem V. V. (BVVu) - f
Ou
u = 0--~ - 0
in ~2,
(6.6.1)
on O~.
(6.6.2)
Here f~ C I~2, v denotes the outward unit normal to the boundary, and B - "[bijkt} E s 2 1 5 We assume that bijkt -- bjikt = bktij,
o,~ 1,71~ _< B~'r] _< ~lvl 2 , v,7 e M~,,x~,
(6.6.3)
and
.f e L 2 (~),
bijk~ e L ~ (gl).
(6.6.4)
To apply the general scheme, we set U - L 2 (f~, M12x2),
V - H 2 (f~),
Ow
v0-{w e v I w - 0v = ~ on0~}, and define A as the Hessian operator. Now, (6.1.1) has the form
BVVu"
V V w dx = / f w
n
dx
Vw E Vo.
(6.6.5)
n
By B -1 we denote the inverse tensor, which satisfies the double inequality
~-~1,7"1 ~ < B-",7* ",7" _< ~-'1,7"1 ~, V,7* e M~ x~, The spaces Y and Y* are equipped with norms
IIIy II!~- f By" y dx;
Uly* 1112.-f B -~ y*. y* dx,
(l, w) - - / f w dx, f~
and
Qt = {y* e Y* l /y* "VVwdx= / fwdx, vwe Vo}. ~2
~2
(6.6.6)
6.6.
ELLIPTIC EQUATIONS
155
OF THE FOURTH ORDER
Since
IlVVwll > a3JJwJJ2,2,n Vw ~ go, we see that the condition (6.1.3) is satisfied. Problem (6.6.1) and (6.6.2) is associated with two variational problems. Problem P .
Find u E Vo such that J ( u ) = inf J ( v ) , vE Vo
where
1/
J ( v ) - -~
" VVv dx -
BVVv
f
f w dx.
N
Problem 7)* .
Find p* E Q~ such that I*(p*)-
sup I*(q*),
Vq*EQ'~
where I*(q*) - --~1
f
B
--lq* 9 q, dx.
fl
By Proposition 6.1.1, we obtain two basic relations for deviations:
Ill v v ( ~ -
~)ill ~ + IIIq* -P* 1112,- 2(J(v) - I*(q*)),
(6.6.7)
and [] V V ( v - u)II12 + ]1 q* - P "
Ill~, -
2D(VVv, q * ) =
= f (BVVv 9 VVv + B - a q * 9 q* - 2VVv" q*) dx,
(6.6.8)
which hold for any v E V0 and q* E Q~. From (6.1.13) it follows that
1
IIIvv(~- ~)II1~ (1 + f l ) D ( V V v ,
y*)+
+
1+~
1) ~a~(y 1 ,),
(6.6.9)
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
156 where d 2 ( y * ) -
inf
q*eO~
Ill q*-y* III2, Note that
f y* 9VVw dx -
f(div
ft
f~
div y*)w dx,
VwE V0,
so that A* : Y* --+ H-2(f~) is the operator div div. Next, (l + A'y*, w) -
_ / ~ ) dx
vw
(6.6.10)
f~
and, therefore,
f(y* : VVw - fw)dx d~(y*) - I I + A'y* I - sup ~ e Vo
III V V w
III
If y* e H(divdiv, f ~ ) ' - {y* e L2(f~,M~ •
I divdivy* e L2(f~)},
then this quantity is estimated by the relation II + A'y* I < sup - ~
Vo
II div div y* - fll~llwll~ III V V w
III
< -
sup II div div y* - fli. Ilwll. < C~. II div div y* - fll., -~Vo ~llVVwll - ~ <
(6.6.11)
in which Cla is a constant in the inequality Vw E Vo.
Ilwlla __ ClallVVwlla
(6.6.12)
By (6.6.9), we obtain 1
Ill V V ( v - u)[112_< (1 + 13)D(VVv, y*)+ +
+
~l[divdiv
-
fii~, (6.6.13)
In this estimate, y* is an arbitrary tensor-valued function from H(div div, f~) and 13 is a positive real number. However, this is rather demanding in relation to the dual variable y* (which must have square summable div div).
6. 7. RELATIONSHIP WITH OTHER METHODS
157
To avoid technical difficulties that rises from this condition, we estimate the negative norm as follows: [l + A'y*[ < C2---q[Idivy* - rl*[[n + Via II divB* - f[[a. ~1
(6.6.14)
Oll
Here, 77* is an arbitrary vector-valued function from H(div, fl) and Cln is a constant in the inequality [IVw[[a _< Cla[IVVw[[a
Vw
6
V0.
(6.6.15)
Then, we arrive at the estimate 1
[[[ V V ( v - u)[[[2_< (1 + 13)D(VVv, y*)+ -[-
(1 -[-
~
(C2f~[] div
9
- ~ I[a + Cla[I dive* - f]]f~)2
(6.6.16)
in which y* 6 L 2 ( f ~ , ~ • must have square summable divergence and ~* 6 H(div, f~). Note that Cln <_ CnC2n, where Ca is a constant in the Friederichs inequality. In view of this, we obtain a slightly different form of the deviation estimate (cf. [149]): 1
III v v ( ~ -
~)II1~< (1 + 13)D(VVv, y*)+
+ (1 + ~) ~C~a (lldiv y* - o*lla + Calldiv rl* - flla) 2 , (6.6.17) For boundary conditions of other types, the deviation majorants can be derived by arguments similar to those used in Section 4. Lower estimates follow from (6.2.4). We have 1
III VV(v- w)1112>Me(V,W)
w 6 Vo,
(6.6.18)
where 1
Me(v, w).= - ~ III VVw III2 -
/
( B v v v . V V w - fw)dx.
f~
6.7
Relationship with other methods
The majorant ,~4e(V, t3, y*) involves an arbitrary function y*. One way of finding a suitable y* has been discussed. It leads to an auxiliary finitedimensional problem (6.3.3). However, one can also define y* by attracting
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
158
the information encompassed in the approximate solution v. In this section, we are focused on this way. Our aim is to show that such modus operandi leads to a wide spectrum of a posteriori error estimates. For finite element approximations, some of them coincide with a posteriori estimates earlier discussed in Chapter 4. Consider the abstract problem presented in w of this chapter. Assume that (/, w) = (g, w), Vw e V, (6.7.1) where g E U. In this case, p* e Q * :={y* EY* IA*y* e U }
(6.7.2)
and, moreover, p*EQ~:={y*eQ*I(A*y*+g,w)=0,
VweV0}.
(6.7.3)
Let
y~ - AAv.
(6.7.4)
y* - ny~,
(6.7.5)
Setting where II is a certain continuous mapping, we obtain a function y* that can be substituted into the majorant Ad~. 6.7.1
Residual
based
estimates
The simplest way is to set y* - y;,
(6.7.6)
which means that II is the identity mapping of Y* to itself. In this case,
D(Av, y~) = 0 and ,/Via(v, ~, y[) contains only the second term, which upon minimization with respect to/3 gives the first form of the majorant 1
A*
,
which implies the estimate Ig A(v - u) Ill< ]l + A*AAv] - sup (g' w) + (AAv, Aw) ~ Vo iU h ~ iil '
(6.7.8)
If v is obtained by the finite element method (so that v coincides with a Galerkin approximation Uh E Vh := Yoh + Uo), then the right-hand side of (6.7.8) is estimated by the method presented in Chapter 4. As a result, we obtain residual type a posteriori error estimates that involve integral terms associated with finite elements and interelement jumps.
6.7. RELATIONSHIP WITH OTHER METHODS
V
. . . . .
159
L
V " "'-.......
y~
0~
-..
J Figure 6.7.1' Various choices of the dual variable.
6.7.2
E s t i m a t e s b a s e d on the "regularization" dual variable
of t h e
Obviously, the best choice of y* in jM~(v,~,y*) is y* = p* E Q* (cf. 6.7.3). Therefore, if y~ ~ Q* and we know an inexpensive continuous mapping from Y* to Q*, then the image of y~ obtained by such a mapping could be a better approximation of p*. We call such a post-processing procedure the regularization of the dual variable and denote the respective operator by IIreg. By IIreg, we define another element. This situation is presented in Fig. 6.7.1, which is taken from [172]. y~ -- l'IregY~ E Q*
(6.7.9)
and the quantity A4, (v, ~, y{), which gives the second form of the majorant A4~)(v)-
inf
{(1-t-~)D(Av, IIreg(v4Av))-}-
BER+
+
(
1+
~l/+
IIreg(c4Av) }.
(6.7.10)
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
160 Now
]/+A*II~eg(AAv)[- sup (g + A*y~,,w)
weyo
III Aw II]
<- C3111g + A*y~ Jl.
Assume that v - Uh, where Uh is a Galerkin approximation computed for a finite element space Voh C Vo and IIreg - G, where G is a "A-averaging" operator (cf. IV.4). Then 1
D(Av, Hreg(Ahv)) - -~ IIIAAuh -G(.AAUh)lit2,
(6.7.11)
and by (6.7.10) we derive the estimate ]Hih ( u h - u)]i]2__ (1 + f~)I][ A A u h - G(AAuh)tit 2 +
+
( 1+ 1 )~
C~2fIg+
A*G(AAuh)]I
, (6.7.12)
whose right-hand side is easily computable. If we set T - AA, then the first part of (6.7.12) is precisely the error indicator (4.4.3). Of interest is to discuss conditions that make this term dominant. Let the regularity of u and properties of approximation spaces provide that mA(uh - u)]11,~ Ch '~, (6.7.13) where a constant C does not depend on h. Assume that the operator G possesses a superconvergence property (see IV.4), i.e Ill A A u - GAAuh Kit,_ Ch'~+~, ~ > 0. (6.7.14) In this case, the majorant (6.7.10) consists of two terms that have different asymptotic rates. Indeed,
(g + A*GAAuh, w) - (GAAuh - ,AAu, Aw) and we find that
I I + h*GAhuh~ ___titA h u - GAhuh m,_< Ch'~+~. We see that A/[~ ) (Uh) tends to zero as h a, but its second term has a higher convergence rate. Thus, the first term given by (6.7.11) dominates, provided that h is sufficiently small and we conclude that
./~(~) (Uh) '~ D(huh, GAAuh).
(6.7.15)
The quantity D(Auh, GAAuh) is easily computable. Therefore, (6.7.15) gives rise to various practically attractive error indicators.
6. 7. R E L A T I O N S H I P W I T H O T H E R M E T H O D S
6.7.3
Estimates
based
on
the
161
"equilibration"
of the
dual variable Finally, consider the case in which the operator H maps Y* to the set Q}* c Q*. In this case, we define y: = Hy; e Q~.
(6.7.16)
Hence, (A*y~ + l, w) - 0,
Vw E V0,
so that [A*y~ + l] - 0. Now the majorant contains only one term: Ad~ ) (v)
-
(6.7.17)
D(Av, y~).
The meaning of this type error majorants is easy to explain by the paradigm of the linear elasticity problem (see VI.5). In this problem, y* presents a stress field, and the set Q~ consists of the tensor-valued functions satisfying the equilibrium equation (y* " e(w) + f . w ) d x + / g~
F . w ds = O,
Vw E
Vo.
(6.7.18)
02 f~
For this reason, it is natural to call II an equilibration operator and denote it by Heq. The respective majorant has the form
1/(: (v)-
34(~)(v)= ~
Ileq(Le(v)))" (lLe(v)-Ileq(Le(v)))dx. (6.7.19)
f~
Let T*(v) = Le(v) be the stress function associated with :(v). Then, we rewrite ~4~ ) as follows"
lfL-:(r*
:
( v ) - n~qr* (V))'(T* (V)- IIeqT* (v))dx -
~2
1 = 5 Ill T*(V)- H~qT*(V)1112,,
(6.7.20)
The relations (6.7.19) and (6.7.20) show that if a computed stress field T* is post-processed in such a way that it satisfies (6.7.18), then the majorant is given by the error in the Hooke's law, or, what is equivalent, by the U[" I]].norm of the difference between r* and its post-processed image IIeqr*. We
162
C H A P T E R 6. L I N E A R ELLIPTIC P R O B L E M S
recall that this type error estimators for finite element approximations have been discussed in Chapter 4. In general, it is rather difficult to construct an operator IIeq with the More-mentioned projection properties. Instead, one may try to use an operator IIeq, which provides an approximate equilibration. In thise case, the second term of the majorant .h4~(V,~,IIeqT*) does not vanish and should be taken into account. 6.7.4
A priori
projection
type
error
estimates
Now, our goal is to show that classical a priori projection type error estimates also follow from the general deviation estimate (6.2.3). Let Uh E Vh be a Galerkin approximation of u. We have
]]]h(u--uh) ]0]___2(1--F.~)D(huh, y*)~- (1-t-~) ]A'y* +/0(~6.7.21) Set here y* - .Ahvh, where Vh is an arbitrary element of Vh. Then, i A'y* + l[ -
sup (y* - p*' hw) = we Vo ]][ hw ]]] = sup e v0
(AA(vh - u), Aw) JiBh w lil
< III A (~ h - ~) III.
It is easy to see that (6.7.22)
D(Auh,.AAvh) - J(vh) - J(uh). Indeed, 1
1
D(Auh, AAvh) = -~(AAvh, AVh) + (l, vh) - -~(AAuh, AUh) - (l, uh)+ + (AAuh, h ( u h - Vh))+ (l, uh --Vh). Since Uh E Vh is a Galerkin approximation, the last two terms vanish and we arrive at (6.7.22). Recalling (6.1.10), we find that 6]]A ( u h - u)I]]2= 2(J(uh) - J(u)),
161A(vh -u)I]62- 2(J(vh) - J(u)). Therefore, 2D(Auh,
AAvh)
-
2(J(vh)
- J(u))
- 2(J(uh)
- J(u))
-
=Ui h ( ~ h - ~)iil ~ -ill h ( ~ h - ~)ill ~
6.8.
163
INDETERMINACYINTHEDATA
Now, (6.7.21) leads to the estimate
(2 + f~)III A(u- Uh)III~_ (1+,6) Ill A(vh- u)II~+ (1+ 1~ p), Ill A(vh-~)Ill ~, \ which shows that
III A ( u -
Uh)II1~_<
(1
1 )
+ /~(2 + Z)
Ill h(~ - ,h)Ill ~ 9
Since/3 is an arbitrary positive number, we arrive at the projection type error estimate (cf. 2.3.14) [1[A ( u
-
Uh) ][I< --
inf
vhEVh
Ill A(u - Vh) Ul.
(6.7.23)
Finally, we note that the general deviation estimate (6.2.3) also implies another projection type error estimate. Let us set v = Uh, y * -- Yh* o= ,AVUh and use (6.1.13). Since D ( A u h , y~) - O,
we have
iii h (~ h - ~) Ul: _< lil yh* - q* iU,~ Vq* e Q r. From here, it follows the estimate ]]] A(u - Uh)]]] < --
inf
q*EQ~
]]] y~ -- q* ]lJ,,
(6.7.24)
which is in a sense dual to (6.7.23). It shows that an upper bound of the error is given by the distance (in the space Y*) between y~ (which is a dual counterpart of the Galerkin approximation Uh) and the set Q~ containing the exact solution p* of Problem 7~*.
6.8
Error
estimates
determinacy 6.8.1
General
taking in the
account
of the
in-
data
concept
In the majority of applied problems, exact values of the data are unknown. For example, in linear elasticity problems material constants may be known
164
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
only approximately, so that instead of exact numbers we have some intervals. This indeterminacy must be taken into account in the error analysis especially if the influence of the arising additional errors is significant. Below we consider this question by the paradigm of the problem A*AAu = l in ~,
u = uo on Off.
(6.8.1)
where the operator Jt and the functional l are defined with some indeterminacy. Their actual values are unknown. Instead, we know the sets of admissible data. Formally, this means that A E/~A C L:(U, U),
(6.8.2)
l E/it C Vo*,
(6.8.3)
where/~A and/~z are bounded and connected sets that take into account possible variations of the data. It is easily observed that the majorant M s and the minorant M e derived in Section 6.2 explicitly depend on A and I. Because of this fact, functional type majorants and minorants can serve as numerical tools naturally adopted to error estimation if data of a boundary-value problem are not exactly determined. Below, we show how such an estimate can be obtained. To make the exposition clearer, we demonstrate this with the paradigm of the problem (6.4.4)-(6.4.5). Possible generalizations of these results to other linear elliptic equations are rather obvious. In essence, they require choosing proper spaces V and Y and finding the constant C(fl, A), which also depends on properties of the operator A. First note that the solutions of problems with above data form the set T(U~t, Ut) " - { ~ E
Vo + u o I
satisfies (6.8.1) for some~ ,4 EhCA and l E/4t S
Let v E Vo+uo be an approximation of an unknown exact solution. Since the data are indeterminate, the error estimation problem comes in two different forms. The first problem is to find the quantity 2i n em
(v,T)-
inf ~1 [[A(v -
SET
u)l[ 2
(6.8.4)
where the multiplier 1/2 is present for convenience. The quantity e m i n measures the distance between v and the set T. It is equal to zero if v satisfies (6.8.1) for some pair (A,l) E /~A x / ~ . This quantity provides a lower bound of the true error or the error in the best-case situation when the exact solution is the function of T closest to v.
6.8. I N D E T E R M I N A C Y I N T H E D A T A
165
Another problem is to find the quantity 2 (v T ) - s u p ~1 IIA(v-~)] [2 , emax ~ET ,
(6.8.5)
which shows an upper bound of the error. It takes into account computational errors and errors caused by indeterminacy and shows the error in the worst-case situation when the exact solution is an element of T that is most distant of v. This quantity is always positive and its value gives an idea of the accuracy limit dictated by the effect of indeterminacy in the data. Thus,
emin(V,T) <~_e(v) ~__emax(v,T))
(6.8.6)
where the actual error e(v) is principally unknown and we may only hope to find its bounds. In general, the exact values of emin and emax could hardly be found. However, using functional type a posteriori estimates, one can find their computable bounds. Indeed, the majorant jt4~ and the minorant M e explicitly depend on ,4 and l, which opens a way for computing errors caused by the indeterminacy in values of the problem data. Below we show how such an account can be performed. Assume that the set T is known. Our aim is to find practically computable numbers ee(v , T) and e~(v, T) such that for any v E V0 + u0 the following relations hold:
ee(v , T) ~ emin(V,T) < emax(V,T) < e~)(V,T).
(6.8.7)
These numbers give bounds of the total error that is caused by the computational inaccuracy and data indeterminacy. It is worth noting that such type estimates can also be applied to differential equations with random coefficients.
6.8.2
U p p e r b o u n d of t h e
error
In the subsequent text, we take the problem (6.4.4)-(6.4.5) as a basic example. In this case, V = H I (12), V0* = H-l(i2),
Y
=
L2 (fl, I~n),
Av=Vv,
. - ~ ~(fl), Vo.-
A*y*=-divy*,
and ,4 is a mapping given by the relation y*(x) --+ A(x)y*(x), where A(x) is a symmetric positive definite matrix.
166
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
Assume that the indeterminacy in the coefficients of the differential equation (6.8.1) is described by fixing some "mean" elements Ao E L~(fl;1M~ •
and
l0 E L2(~)
and defining bounds of possible variations. In this case, the elements of the sets HA and/~t are represented by elements of the following two functional sets:
HA := {A E L ~ ( g ~ ; M ~ • H I "-- { f E L ~ ( ~ ) i f -
Ao + r
E E E},
fo + (f~o, ~o E .7:},
where g' := {E E L ~176 (~; h~s xn) I [I IEI II~,n .T" := {~o E L2(~)
_ 1},
111~112,~_< 1}.
Here, ~ and 5 are small parameters characterizing the range of indeterminacy. Henceforth, we assume that the parameter ~ is small enough, so that the problems remain uniformly elliptic for all possible data. Formally, this means that the condition cxI~l 2 < Ao~-~ < v2I~I2,
v~ e ~",
(6.8.8)
implies similar estimates for all A C/4A. Since IE~. ~l _ IEI IS[2, we find that A~. ~ > Ao~. ~ - ~IEI I~12 _ (cx - ~)1~12, A~'. ~" < Ao~. ~"+ ~IEI I~12 _< (c2 + ~)1~12.
(6.8.9) (6.8.10)
From here, it is clear that e must be subject to the condition < cl.
(6.8.11)
For the inverse matrix, we have c~lI~[ 2 _ Aol~ 9~ _ c~-1]~[2, (c2 + ~)-1 ]~[2 ___A - I ~ . ~ _ (cl - ~)-1 i~[2, where A E/4,4. To use (6.4.8), we must estimate the functional 1 (A -~ A'y* , A'y* ) - ( A v ,y* I(AAv, A v ) + -~ ) D(Av, y*) "- -~
(6.8.12) (6.8.13)
6.8. I N D E T E R M I N A C Y I N
167
THEDATA
for any A E Ha. Denote the unit matrix by IT. Then A -1 - (Ao(lI + r
-1 - (]I + r
-1 A o 1.
Note that
~IAolE{ < ~llAoXl IEI < ~c-~~ < 1, and, therefore, (]I + r
where B - A o l E . Hence,
-1 = I[ + ~ (-1)Jr j=l
A - l y *. y* - (I + r
* . y* = oo
= A o l y * 9 y* _ e B A o l y * . y* + E ( - 1 ) J e J B J A o l y
*. y*.
j'-2
Since E E t;, we have BJAoly
. y* dx <_
n
/IAo~{ j+l IEI~ ly*l 2 dx
<_ c~-(w+~)lly*ll 2
f~
and A - l y , . y, dx < _ / ( A o l y *. y* - e B A o l y * . y*) dx+
ft _b (~(_l)J~JCl(J+l)) " j = 2 ,y,2 Moreover, A V v . V v dx - / ( A o V v .
ft
V v + e E V v . Vv)dx.
ft
From these relations, we find that the first term of the duality error majorant explicitly depends on ~. It is estimated as follows" D ( V v , y*) <_ D0(Vv, y*) + c / ( E V v .
V v - B A o l y *. y*) dx+
fl
1
y , i 2,
e j=2
j
(6.8.14)
168
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
where D o ( V v , y*) -
/(1 -~AoVv.
1
V v + -~Aoly* . y
,-
Vv.
y,)
dx.
f~
Since all the matrices are symmetric, we have E V v . V v - A o l E A o l y * . y* - E ( V v - A o i y * ) 9 (Vv + A o l y * ) .
Now, (6.8.14) implies the estimate D ( V v , y*) <_ Do(Vv, y*) + ~
~f
E ( V v - A o l y ). (Vv + A o l y ) d x +
*
e cl
9
1 ,i 2 2(6+cl)11Y I 9 (6.8.15)
Recall that (see (6.3.17)
1 III V ( v - u ) I I I 2 -
inf
y*EQ* f~>O
.M$(v,~,y*).
(6.8.16)
For the problem considered, we have UlV(v - u)III 2 -
f(Ao
+ ~E)V(v - u). V(v - u ) d x >_
f~
>_ ( ~
-
~)IIV(v
-
u)ll ~ .
(6.8.17)
Therefore, 2 emax
( v , T ) - s u p l ~ IIV(V -- ~)II 2 < 1 sup III V(v - ~) II1~= JeT -- 2(Cl -- e) ~eT 1 = sup inf M e (v, f~, y*). (c~ - ~) a c t y" eQ" ~>o
By definition, T (b/A,//f) is a subset of V0 + uo that contains solutions of the boundary-value problems (6.8.1) with A-
Ao + ~E
and
l-.fo+&p.
6.8. INDETERMINACY IN THE DATA
169
Since any such problem has a unique solution, we can replace sup by sup. (A,f)
Hence, we obtain CI
-
-
g) emax(V 2 ,T) _ sup
inf ~4r y*) _< ~eT Y*~Q* 3>o < inf sup ~ / r (v, 3, y*)--- y*EQ* fiET f~>o
=
A ) 2(~' /3 l+f~)D(Vv, y*)+ (1 +f~)C2
inf. sup
]Idiv Y* - f ] l 2}"
y* EQ AEI,4A
From here, we deduce the main estimate
e2m~(v, T) __ 1
(1 + 3) sup D(Vv, y*)+
1+
C1
AEblA
Cl -- C
%.
+ (1 +/3) sup C 2(~2, A) sup i] div y* - .fi]2
23
.fEll/
AElgA
(6.8.18) J '
which is valid for any y* E Q* and f~ > 0. The first term on the right-hand side in (6.8.18) is estimated by (6.8.15). Let us consider the second one. We have
f AVw. Vw dx
1
= inf ~ C2 (~2, A) ~eVo
Ilwll2
where
AVw . Vw dx > / A o V w . Vw dx - eiiVwl[ 2. Hence,
1
f AoVw" Vw dx
C2(D,A ) > ( 1 - e c ~ -1) inf ~ -
Cl
-
-
cl
II ll
C~(~,Ao)
and
e
C 2 (gl, A) _ (1 + el
-
-
) C2(gl, Ao).
(6.8.19)
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
170 Now, we see that {
sup
AElgA
(1 + 3)D(Vv, y*) +
(1) 1+ ~
C2(f~' A) y, 12} 2 lldiv - fl <
feus
< (1 +/3) Do(Vv, y*) +
6
1 2(e c~-------------~ + Ily*II2§
6 sup f E(Vv - Aoly*) 9(Vv + Aoly *) dx) +
+ 2 Eeg
n
+ ( 1+
6 ) (l+13)C2(f~'A0, sup cl - 6 23 ~e~
f
Idiv
,,
- f o - 6 q o dx. (6.8.20)
n
Note that for any g E LZ(f~) sup f ( g - r ~o6.Y"
- {[gll2 + 2{Igl[+ 1.
t2
By this relation, we find the value of the very last term of (6.8.20). Recalling (6.8.19), we arrive at the estimate
emax
(,, T) <
< - -1 ( 1+ -cl
6 ) {inf ] 2 c l cl'j (1+]3) ( Do(Vv, y')+ ( e[--'l)y *2]2(6+ cl-e v'eQ" p>0
i(Vv - Aoly*) 9(Vv +
+ ~
Ao~y*)ld~)+
f~
6 ) (1+13)C2(~,A0) 2/~
+ ( 1 + ~Cl --6
(lldivy*-foll2 +2~lldivy*-foll+82)} 9 (6.s.21)
To represent the right-hand side of this estimate in a more transparent form,
6.8. INDETERMINACYINTHEDATA
171
we introduce the following quantities:
Moo(v, fl, y* ) := (1 + fl)Do(Vv, y*) 44- (14-~_1) C2(fl, Ao) ll divy, _ M ~ o ( v , ~ , V*)
8 /
5(
=
I(w - Ao
ly,
) ( w + Ao y*)l
+
f~
+
(1) C (n, Ao, f ldivv, - ,ol 1+~
cl-s
fl
Mol (v, fl, y*)
9= 6 ( 1 + ~ 1 ) C 2 ( ~ , A o ) l l d i v y . _ f o ,
M22(v, fl, y*)
C2(f~,AO)lldivy._
(,1)
Mll(v, fl, y*)
-
e
:=
(1 + ~) ~-i-
+
( 1 ) 1+?
8
Ily*ll 2
2(s + ~) +
clC2(f~,Ao) 62 2(c1-~)
Now, we see the structure of the upper bound
4(~, T) _-
1
C1
I
"
ee(v, T) in (6.8.7):
-
1 4-
el
inf -- 8
y* E Q *
/5>0
Mst (v, fl, y ) 4-/1//22(v,/3, y*)
.
(6.8.22)
s,t=l
The term Moo(V, fl, y*) does not contain the small parameters ~ and e. It coincides with the majorant constructed for the "mean" problem (A - Ao, I - /Co) and represents the major part of the approximation error. The terms Mlo, Mol, and Mll are given by some combinations of the weighted residual and small parameters ~ and ~. In principle, all these terms can be made arbitrarily small by taking v close enough to the exact solution u of the problem with A = A0 and l = f0 and y* close enough to AoVu. In contrast, the term M22(v, fl; y*) is always positive. This term contains the inherent part of the error, which does not depend on the accuracy of numerical approximations. Indeed, in all cases we have
01~2
M~2(v,P, y') _ eo "- O~(~, Ao)2(~ - ~)"
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
172
This quantity does not depend on the choice of v, /3, and y*. It gives an idea of the accuracy limit that could be achieved within the framework of the worst-case scenario. For example, if A0 is a unit matrix and fl = (0.1) x (0.1), then Cl 1 and C2(ft, Ao) = 1/27r 2. Assume that the coefficients are defined with the ambiguity rate c = 1% and l with 6 = 2%. In this case, 00 ~ 10 -5. -
-
"
A series of computable upper bounds is obtained if we replace Q* by a sequence of finite-dimensional subspaces {Q~} c Q*. Then, we have
e~(v, T) <_e2ke(V, T) -
1(1
= -C1
+
inf 01--~' y*EQ~ /~>0
}
Mst (v,/3, y* ) + M22 (v, fl, y* ) .
(6.8.23)
s,t=l
If Q~ c Q~+I, then the sequence {e~e(V, T)} monotonically decreases but may not tend to zero.
6.8.3
L o w e r b o u n d o f t h e error
To find a lower bound, we use the relation
I / AV(v
2
u) V(v
u) dx - sup ~4e(V,W), wEv0
fl
where Me(V,W)--/(~AVw.Vw+AVv.
Vw+.fw)dx.
f~ Recall that A~. ~ _ (c + c2)I~12. Then, by (6.8.4), we see that for any (A, f ) E ~4fA•162 and the respective solution ~, the following relation holds" 1
IIV(
-
)11
2
1
>- 2(r + c2) =
1 ~+c2
/ AV (v - ~). V (v - ft)dx = f~
sup Me(V, w). wEVo
6.8.
INDETERMINACY
173
IN THE DATA
Therefore, emin(V' T ) -
ainf e T ~1 IIV(v - a)l
1
>
12 >__ inf
sup
. M e ( V , w) >
-- 6 - } - C 2 (A,S)6blA xbll w6Vo
1
>
sup
inf
-- r -~- C2 w6Vo ( A , S ) 6 U A
M e (v, w) x/J'i
{-/ (1-
-_
~ A o V w . V w + A o V v . V w + low
r +1c2 -e-o'U'
)
dx+
n
+ inf
EEE ~6 Yz
( S( 1 -e
EVw.
Vw+EVv.
Vw
2
~
)
dx-6
S )} wqodx
(6.8.24) Here,
inf{-S(2EVw'Vw+EVv'Vw) dx} -
E6 E
fl
--li( 1 n
and
fw~dx}--Ilwll.
inf { -
~6.T"
n
Now, we obtain 2 emin(V, T) _>
>
1
sup { - i ( ~
-- 6-t-c2 wE Vo
)
AoVw'Vw+AoVv'Vw+fow
f~
-e
/I' l
I NVw + Vv | Vw[ dx
fl
-
allwll
dx-
}
9 (6.8.25)
174
C H A P T E R 6. L I N E A R ELLIPTIC P R O B L E M S
Introduce the quantities
moo(v,w)
=
-
/(1
-~AoVw . Vw + A o V v . V w + fow
)
dx,
n
-
,
mo,(w) =
-,~ll~ll.
Then, we represent the lower bound in the form
e~(v, T) :=
1
sup {mo0(v, w) + m01 (v, w) + mlo(w)} >_ 0.
(6.8.26)
wEVo
g + c 2
In this estimate, the term moo (v, w) contains the major part of the approximation error. It vanishes if v is a solution of the "mean" problem with A - Ao and l - fo. Two other terms reflect the influence of the small parameters 5 and e. To obtain a computable lower bound, we replace V0 by a finite-dimensional subspace Vok and solve a finite-dimensional problem ~ (v, T) >_ ~ e ( ~ emin
,
T) -
1
~up {moo(~,~) + mo~(,, ~) + m~o(~)}.
g + c2 wEVo~
The quantities eke2(v, T) show the efficiency of further computational efforts within the framework of the best-case scenario. If they are large, then approximation errors are significant, and using a more precise method of approximation is sensible. On the contrary, if all these quantities are small, then an approximate solution found is probably close to a function u E T, so that further mesh refinements do not give much additional information.
6.9
Error estimation
in t e r m s
of linear func-
tionals In this section, we apply two-sided error estimates given by ~ 4 e (or Me) and M e to deriving sharp bounds of errors evaluated in terms of a linear functional L Consider the problem (6.1.1) with I = f and another problem that is to find v E V0 + u0 such that
(AAv, Aw) + (~, w) = 0,
Vw E V0,
(6.9.1)
6.9. LINEAR FUNCTIONALS
175
where s is a given functional in Vo*. By Proposition 4.5.1, we have
(6.9.2)
where ~ is an approximation of u, ~ is an approximation of v, and E0 (~, ~) = (f, v-') - ( A A ~ , A~),
(6.9.3)
E~ (~, ~) = (AASf, ASt),
(6.9.4)
5f = ( u - ~), 5~ = ( v - ~).
Two-sided bounds of the term (AASI, A51) can be found by two-sided energy estimates as follows. Note that for any positive a we have 2(AASt, ASf) = 1 = Ill A (aSf + 15e) Ill2 - a 2 Ill A~f [[]2 _ ~ Ill A~e Ill2.
(6.9.5)
By the majorant M e (or M e ) and the minorant M e , we find two-sided estimates of the energy norms of 51 and 5t. Let y~, y~,/3f, fit, Wl, and wt be the functions and parameters that correspond to these estimates, i.e, 1
mY := .Me(~,ws) < ~ Ill A'b Ill~< .,~e(~,Z,,,y.;)"- M s, 1
m, . - .Me(~', w,) < E III ASl II1~< .Me(~',,~,,y;)"-
Me.
In this case, the quantity
III A(o,~ +
,Se)i11~-IIIA ( a u
1
+ -v-
1
a~
"6) lU~
(6.9.6)
can be viewed as the norm of the difference between the exact solution 1 u~l E Vo + (a + ~)uo of the problem
(g A u l l , Aw ) + (oL.f + l e, w) - O ,
Vw e Vo
(6.9.7)
and the function v~l - a ~ + ~v 1 - E Vo + (a + ~)uo, 1 which is an approximation of u~l. To obtain two-sided bounds of the norm (6.9.6) we use already known functions y~, Y'l, wf, and w~ and compute the numbers ~ I ~ = Me(,,?~, ~ I
+
1 ~), I
,
M ~ - M e (v~, ~, ~y'j + -y~ ). Ol
176
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
We note that ml, e = mfe(a), Mfe = M fe(a, ~) and positive numbers a and /3 can be taken arbitrary. By (6.9.5), we obtain (AhSf,hSe) < -
me a2mf - ~-~-} := ff)~$,
inf { M f e a,flER+
(Ai(f/,, iSe) >_ sup { m f e - a 2 M / -
Me
--~-} := ff)~o.
aER+
Recalling (6.9.2), we now deduce a two-sided estimate
+ E0( ,vD <
( e, u -
<
+ Eo (
v-) ,
(6.9.8)
Note that if y~, y~, wf, and we provide accurate two-sided estimates of the energy error norms in the problems (6.1.1) and (6.9.1), then Mf e and m f e furnish accurate two-sided estimates in the problem (6.9.7) and, consequently, (6.9.8) yields sharp upper and lower bounds of (g, u - u/.
6.10
Practical
implementation
In this section, we analyze the effect of duality error majorants (DEM) and compare them with the results obtained by other methods. 6.10.1
General
scheme
Two-sided estimates obtained in Section 6.2 suggest a general algorithm for solving Problem P with any a priori given accuracy A. First we choose a set of finite-dimensional spaces {Irk } C V (dimVk = Ark ~ +
I Choose a desired tolerance A, set k = 1, s = 1, and define Vk and
Yr. S t e p 2 Find Uk by solving Problem ~ on the space Vk. Step
3
Find M~ s
-
inf M~ (Uk,/3, y*). y'EYe*
6.10. P R A C T I C A L I M P L E M E N T A T I O N
177
S t e p 4 If M~ 8 < A, then go to Step 8. S t e p 5 Find Ad~ -
sup .h4o(Uk,W). wEv~+l
S t e p 6 If Ad~ > A, then replace k by k + 1, Vk by a refined space Vk+l and go to Step 2. S t e p 7 Replace s by s + 1, Ys* by a refined space Ys*+l and go to Step 3. S t e p 8 Print Uk and stop. The first step defines spaces originally used for obtaining an approximate solution (Step 2) and finding the first (rough) upper estimate of the error (Step 3). If A exceeds the majorant (Step 4), then the desired accuracy is achieved and the process passes to the final Step 8. If it is stated that the minorant exceeds A, then Uk does not have the required accuracy and it is necessary to solve Problem 7~ on a refined subspace Vk+l (Steps 6 and 2). The situation that occurs if
deserves a special comment. In this case, it remains unclear whether or not Uk is close enough to u. To answer this question additional efforts are required. One possibility is to refine Vk and another is to refine Ys*, trying either to find a better approximation or to compute a better estimate for the existing approximation. Since uk -+ u and our two-sided estimates are exact (see Section 3) , this theoretical algorithm will end with finding a proper approximation Uk. However, any particular computer has a certain limited power, so that any problem can be solved only if A _> Ao, where A0 depends both on the problem considered and the computer used. Certainly, the above algorithm is rather schematic and can be viewed only as a skeleton of reliable numerical algorithms to be used in practice. Such algorithms should include numerous improvements focused first of all on accelerating the process. It could happen that these computations are terminated by time limitations. In this case, the very last value of M~ l shows the best accuracy achieved, which gives an idea of the required power of a computer to be used for finding an approximate solution with the desired tolerance A. Finally, it seems worthwhile to add one more remark. Steps 3 and 5 are related to variational problems, so that one may ask about the sensitivity of
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
178
the algorithm with respect to inaccuracy of their solutions. To clarify this point, we recall that M e ( v k , ~ , y*) provides an upper bound of the error for any y* E Y* and Me(vk , w) provides a lower bound for any w E V. For this reason, exact solutions of these variational are, in general, not required. For example, it may occur that on some stage of the minimization procedure the value of M e becomes less than A. Then computations may be terminated regardless of that how close "current" function y* is to the minimizer y*. 6.10.2
Minimization
of the majorant
Consider the majorant M e ( v , ~ , y * ) defined by the relation (6.3.15). A coarse upper bound of the error is easy to compute by setting y* - GAy, where G is an operator that maps Av to Q* (e.g., if v is a finite element approximation, then G is the standard gradient averaging operator). Then, we obtain the error majorant
M e ( v , 3 ,GAy) = (1 + 3)D(Av, G A v ) +
( 1 ) 1+ ~
c2 ,2 -~ll/+ A*GAvl .
A sharper estimate requires a minimization of M e with respect to the variables y* and /3, which can be done by a direct minimization of M e or by finding a minimiser as a solution of the respective the system of linear simultaneous equations. The latter way is considered below. Assume that
v;, ..., v;),
v* e Y : .=
where y~* E Y* are given trial functions. Then any y* can be represented in the form
k (6.10.1)
j--1 Define the matrix M ~ with entries
* m~t - ( 1 + ~)
C2
(A-ly~,yt) + ~(A*y*,h*y;)
and the vector F ~ with components F ] = (1 +fl)
( (Av, y ; ) - ~(A*y~,/) c2 ) .
/ ,
s,t = 1,/
179
6.10. P R A C T I C A L I M P L E M E N T A T I O N Then
Mr
= (~(~,7) "- -~M 7 "7-- F~ "7 + #~,
where _ (1 + j3)
C2
12
and 7 = {%}In view of the property of M s , we have 1
III A(v - u)
III
inf 0(/3,7). BeR+, ~Y:
(6.10.2)
The problem on the right-hand side in (6.10.2) can be solved with help of a direct minimization method that creates a sequence {/3j,7j} as follows: (A) Find f/j E ~ such that (I)(/3j, 7j) = inf r
7j) := MS
and (B) Find 7j+l E Ys* such that ~(/3j, 7j+1) = inf (h(f~j,7). 7~Y: This algorithm creates a decreasing sequence of numbers M S tending to an upper bound of the error. Each number is a reliable upper bound of the error, so that if MS _ A, then the process may be terminated. Remark 6.10.1. Minimization of the majorant can be performed by the least-square methods developed for some classes of elliptic problems (see, e.g., [33]). Also, we note that the minimization problem can be partitioned into several subproblems associated with some blocks of 7. Therefore, one can accelerate the minimization process by various parallelization procedures. Remark 6.10.2. Let the functions y~ be properly chosen and M E has an inverse matrix for any positive/~. Then, (~(/~,7) is minimized by the vector
and the respective minimal value of this functional is easy to find. In this case, for a given f~ we obtain the estimate 1 Ill h ( , -
_
Ill <
#a
-
1M
1Fa 9 Fa
(6.10.3)
180
6.10.3
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
Effectivity index and
shape
index
In the examples considered below, we evaluate the quality of error estimates by means of two numbers. The first number is the so-called effectivity index /eft :=
Me e
Me - e = 1+ ~ , e
(6.10.4)
where 1 -
Ill h (, -
III
and M e is an estimate of this quantity obtained by an error majorant. If /eft is close to 1, then M e gives a sharp upper bound. The second number characterizes the distribution of subdomain errors. Let 7" be a partition of ~ into m nonoverlapping subdomains ~i, i = 1,2, ...m. For obvious reasons, it is desirable to have an error majorant that not only shows the total value of the error, but also indicates errors in various subdomains. We suggest to evaluate the latter capacity by a special number called the error shape index m
i esh -- 1 + i=1
>__ 1.
(6.10.5)
e
In the above relation, ei is an error associated with ~i and #i denotes an approximate value of this local error computed by the majorant. If i esh is close to 1 then the majorant provides a sharp error estimate and, in addition, the corresponding numbers #i serve as good indicators of subdomain errors. Now we turn to examples.
6.10.4
N o n - G a l e r k i n approximations
Below, we present results of several numerical tests that demonstrate the capability of the D EM error estimation method to correctly estimate errors of approximate solutions regardless of their closeness to the exact one. We begin with a simple 1-D problem (c~(x)u')' = f(x), = 0,
(6.10.6) =
(6.10.7)
It is easy to see that (6.10.6) is the Euler equation for the functional b
a
6.10. P R A C T I C A L I M P L E M E N T A T I O N
181
Assume that
a E L~176
a >_ cl > O,
f C La(a,b).
(6.10.8)
Then, the problem (6.10.6)-(6.10.7) is uniquely solvable. To estimate the errors, we apply the general scheme with U - L 2(a,b),
Y * - L 2(a,b),
o
V0 - H 1(a, b),
Q* - H 1(a, b),
and
Vo + uo - {v e H 1(a, b) I v(a) - O, v(b) = Ub}. In this simple case, A is the differentiation operator A'y* - v* E H -1, where
/
b
y*r
= -
a
/
o
b
Vr ~ H ~ ( a , b ) .
v*r
a
Now, the duality error majorant Mm has the form (cf. (6.3.15)) b
Mm(v,~,y*) - 1 +2~ f l a y ' - y *
[2 dx+
a
b
+ I -;/3 c2(~,b) /[Y*' - f[2 dx.
(6.10.9)
and the minorant M e has the form b
,~(v,
w)=-
/(1
~lw'l = + ~v'w' + / w
)
.
(6.10.10)
a
The exact solution admits the integral representation
u(x) -
/ 1 / ~(t) a
x f (z)dzdt nu -~ a
Ub -
~ a
1
f (z)dzdt
. (6.10.11)
a
In the tests presented below, we consider piecewise affine approximations v computed on regular meshes with N intervals. Our aim is to show that M e and M e provide good error estimates for exact solutions of the respective
182
CHAPTER
6. L I N E A R E L L I P T I C P R O B L E M S
finite-dimensional problems as well as for solutions that do not satisfy the Galerkin orthogonality condition. Therefore, we compute approximations by a direct minimization of J and analyze errors arising on different stages of this process. To find a sharp upper bound, we minimize M s with respect to y* and ~ starting from the function y~ = G(v'). Here, G : L ~ ( a , b) --+ H 1(a, b) is a simple averaging operator. For example, G may be defined by the relations =
1
- 0) + v'
=
+ 0)),
-
Computing inf M s ( v , ~, y~ ), ~>0 we find the first (rough) upper bound of the error. It is further improved by minimizing M s with respect to y*. To compute a lower bound, we define a recovery operator R that maps Y* to Vo + u0: R y * ( x ) :=
a(t) dt + -~
a(t) dt
Ub-
.
It is easy to see that Ry* = u if y* = au'. The quantity M e ( v , R y * - v)
gives a lower bound of the deviation. Let us begin with an elementary example in which a ( x ) = 1, f ( x ) = 2, a - O, b - 1, Ub -- 1, and C(a,b) -- 1 / r 2. In this case, it is easy to find the exact solution u-~
+
1-
x
and compare the actual errors with their estimates. Below we do this for various approximations v. First we take a "very rough" approximation v = x (for c - 0.5 the functions u and v are depicted in Fig. 6.10.1(a). In this simplest case,
1 ~ll(v - u)'l ]2 - ~o 1(1 - 2x)2dx - c2/48 ~ 0.006c 2.
6.10. P R A C T I C A L
IMPLEMENTATION
1.2
183
0.004
n
I
Errors DEM
V U
1
i
0.8
i
.I 0.002
0.6
al~
0.4
a
a a '1
0.2 0
0 0
0.5
1
la
ill% ~a lilil Ill lieD I Ill llll~ ~ll mallei Ilia mmlmmh ~mlm iililll lilli iaalae7 ~lani illlllm~ HNDIII Dllllll ~ IIIIn lmllllll Illlllil mmmmmlmmm I mmmmmmmmm
0
(a)
0.5
1
(b)
Figure 6.10.1: (a) Exact solution and an approximation, (b) distribution of local errors.
For any partition of the interval (0,1), the function v - x can be viewed as a piecewise affine approximation of u. The corresponding subinterval errors e2i for the uniform partition with 20 subintervals are depicted in Fig. 6.10.lb. We compare t h e m with the errors computed by the m a j o r a n t (6.10.9) with different y*. If we set y* = y~ - 1 (cf. (6.7.4)), then the first term vanishes and M e (v, f~, y~') -+ c 2/27r 2 ~ 0.05 c 2
as fl --+ +oc.
Hence, we see t h a t the quantity obtained by the only one (residual) term of M S gives a rough upper bound. However, if we take y* = = c x + 1 - c/2, then the second term vanishes and we obtain another upper estimate: 1 C2
M . ( v , fl , y ~ ) -~ -~
x 0
dx--~
a s f ~ - + O,
184
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
which is exact. Thus, we see that the value of a computed upper bound strongly depends on the function y*. Now let us show that this dependence also takes place when the majorant is used as an indicator of local errors. Namely, if y* is defined in the process of minimization of M e , then the local errors given by the majorant practically coincide with the true distribution (typical picture is presented in Fig. 6.10.1(b). However, attempts to use simplified forms of the majorant that eliminate either the first or the second term may lead to quite different results. In our simple case, this fact is easy to demonstrate. Indeed, the choice y* - y~ - 1 zeroises the first term, so that b
1 +/3c2 / [2 M e (v,/3, y~) 2~ (a,b) If dx. a
In the integral sense, this majorant gives a corect result, but cannot serve as an error indicator, because its integrand given by the residual term is constant (f = 2), while the real error distribution demonstrates strong heterogeneity. It is easy to present even a more striking example showing that the use of averaged derivatives without taking into account the second (residual) term of M e may lead to a confusion. In our case, all sensible averagings of v ~ = 1 give exactly the same function, so that G(v') = G(1) = 1. Therefore, G(v')
-
v'
-
0
and the estimator I[G(v') - v'l[ (which is, in fact, the first term of M e for y* = G(v~)) leads to a wrong conclusion that the error is equal to zero. However, taking into account the second term changes the situation and gives a correct upper bound of the error. This observation shows that error indicators based on comparing the gradients of approximate solutions with their averaged values are not applicable to arbitrary conforming approximations without the second (residual) term. To give further illustrations, we consider the functions u~ = u + ~r where ~ is a number and r is a certain function (perturbation). Approximate solutions (whose errors are measured) are piecewise affine continuous interpolants of u~ defined on a uniform mesh with 20 subintervals. If ~ = 0, then v coincides with the interpolant of u. In the first series of tests, we take r = x sin(zrx) and ~ = 0.1, 0.01, 0.001, and 0. The results are presented in Table 6.10.1. For r = x sin(27rx), the results are exposed in Table 6.10.2.
6.10. P R A C T I C A L I M P L E M E N T A T I O N
185
0.008
Errors DEM
Errors GA
0.006
0.004
0.002
~._+L[
0 0
0.5
0.5
(~)
(b) 1
v!
-
U
0.8
I
I
I
left
-
-
I
~ .
.
.
.
.
.
.
.
.
, . . . . . . .
, . . . . . . . . . . . . . . .
,-
..=
0.6 0.4
-
,,7
,'7
0.2 -
=
I
0.5
i,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
m
I,
,7 o
I
0 0
0.5 (c)
1
I
I
1 3 5 7 9 (d)
Figure 6.10.2" Error estimation for ~ - 0.1.
I
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
186
I
I
I
Errors DEM
Errors GA . . . . . . .
8e-05
! "
I'1
#"" Ii1!!
4e-05
ji,~
Im
IIii
im
"'
0 0
0.5
1
0.5
(a)
(b) I
~
3
1
I
I
left
I
I
----:--
_
1.5 8e-05
_ Error /
,
7" ,
, , ~
4e-05
~
0.5 ~ %
.
,
~
~
~
,
, ,
, ~
, ,
~
,
,
.=..,.,,
,m
,m
0 0.5
(c)
1 3 5 7 9
(d)
Figure 6.10.3" Error estimation for 5 - 0.01.
,,...
,=b
.,.
6.10. P R A C T I C A L IMPLEMENTATION
187
I
I
Errors DEM
Errors GA .
4e-05
.
.
.
.
.
m
2e-05
0
I
i 0.5
0
0.5
1
(a)
(b) I
g
. . . . . . .
10 9~
4e-05
.
.
~
L.
......
!
I
................ , L~S~ ~ .~S S uii ,IDIjlIIII
i,]1
9 . . . . . . . . .
14,1 iii,
beta . . . . . . . ID u I
1.5
[
-~
_
1 2e-05
O5
0
0 0.5 (c)
1
3 5 7 9 (d)
Figure 6.10.4: E r r o r estimation for 5 - 0.001.
188
C H A P T E R 6. L I N E A R ELLIPTIC P R O B L E M S
, -
"
I
" -
,,,
,,
I
Errors DEM
4e-05
,, i,
-
Errors GA
- -
............. .......
iii
...... :11' 1 I!
3e-05 2e-05 le-05
J
i
0
0
0.5
0.5
1
(a)
(b) ! .,,,,,.,m.,.,.,,..m...,,,..
10 -
beta .......
................
20
-
-
-
1.5
m ~ ~
-~m
~
m
.
.
.
.
.
.
.
4e-05
.
.
.
~.
1
3e-05 0.5
2e-05 le-05 I
0
0.5 (c)
0 1 3 5 7 9 (d)
Figure 6.10.5: Error estimation for ~ = 0.
189
6.10. P R A C T I C A L I M P L E M E N T A T I O N
0.02
I
I
Errors GA . . . . . . .
Errors DEM 0.015
0.01
0.005 - ~
I Ill
IL
, i'r4.-_,
0
0
I
0.5
0.5 (a)
0.8
(b)
~
'v ! ---'-- ] u !,
/,/ Z''''
0.6 0.4
I
I
beta
.......
1
I
0
I
I
leff
................!i
0.2 t t 0 ~'t 0 0.250.50.75
I
1.5
., .......
0.5
(c)
'
, .......
, .......
i!
-
i ..........
0
~,~
'.
~
,
I
I
1
3 5 7 9 (d)
Figure 6.10.6: Error estimation for ~ - 0.1.
t
t
190
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
I
Errors
'
I
I
I
Errors GA . . . . . . .
0.0002 -
0.0001
i inll !,11
0 0
0.5
1
0.250.50.75
(a)
(b) I
I
0.0002
10 ................ 20 _Error
I
I
1.5 i
.,. ......................
; ........
I
0.0001
0
l,lllll "=" =. ~'~fi
. m . ,
.
.
.
.
.
.
.
~.
.
.
.
.
.
.
, .
.
.
.
.
.
.
.
.
.
.
.
,-
o
I I I~
0.5 (c)
,==
,
0.5 S~
I
left beta..-------...,._
0
,,,,. .......
I
1
J~ o~..,,p .~, ,~, .~..~, .~, --. -.- -.,,, -.i,~
I
I
3 5 7 9 (d)
Figure 6.10.7: Error estimation for ~ - 0.01.
I
6.10. PRACTICAL IMPLEMENTATION
191
Errors
Errors DEM
GA
----,W j t m
4e-05
ItW flmm I I
u
3e-05 2e-05
IlU
le-05 I
0 0
0.5
0.5
(a)
(b)
2 ...,
m
,..,..,,.,
m
i
m
10 ................ 20 Error
beta . . . . . . . 1.5 1
4e-05 3e-05
0.5
2e-05 le-05
0
0 0.5 (c)
1
3 5 7 9 (d)
Figure 6.10.8: Error estimation for 6 = 0.0001.
192
C H A P T E R 6. LINEAR ELLIPTIC P R O B L E M S
0.1 0.01 0.001 0
e 0.019692 0.001022 0.000835 0.000833
Table 6.10.1: 2M~ 2Me 0.019743 0.019683 0.001025 0.001013 0.000839 0.000827 0.000836 0.000825
left 1.003 1.003 1.005 1.004
iesh 1.018 1.011 1.002 1.002
0.1 0.01 0.001 0
e 0.068365 0.001509 0.000840 0.000833
Table 6.10.2: 2M~ 2.~ G 0.068543 0.068357 0.001520 0.001500 0.000844 0.000832 0.000836 0.000825
ieff 1.003 1.003 1.005 1.004
iesh 1.045 1.042 1.004 1.002
Moreover, we compare the actual values of subinterval errors with the indicators given by the gradient averaging (GA) and duality error majorant (DEM) methods. In the first case, our indicator is a function proportional to ( v ' - G ( v l ) ) 2, where G is an averaging operator that defines nodal values as a simple average of two neighbor values. In the pictures, the values of this indicator are properly scaled in order to fit the actual value of the error. The second indicator is simply the integrand of the majorant M s. Four pictures presented in Fig. 6.10.2 refer to the first line of Table 6.10.1. Picture (c) depicts u and v. True distributions of local errors are shown in (a) and (b) by impulses and compared with the DEM and GA indicators. Picture (d) gives an idea of how the values of the majorant and the parameter ~ change in the process of minimization. Here, the values of the x-axis are the iteration numbers (or time units). In Fig. 6.10.3 these numbers also appear in the picture (c), where they show how the integrand of M s converges to the true error in the process of minimization. Subsequent pictures are associated with other lines of the above tables. We see that, irrespective of the actual appearance of the approximate solution v, the majorant M s yields an accurate estimate of the global norm of the error and also rather accurately reproduces the distribution of local errors. Also, we observe that GA indicator provides a good error indication for small values of ~. Also, we tested the DEM error estimation method for various twodimensional problems. One typical example is presented below. Let u be a
6.10.
PRA CTICAL
IMPLEMENTATION
Table 6.10.3: Estimate left 1.0812 1.477 0.9791 1.334 0.8069 1.102 0.7418 1.013
N 1 3 6 10
193
iesh
2.184 1.790 1.215 1.027
solution of the problem Au = 4x 2 in fl e (0, 1) • (0, 1), 1 lx4
uly=o - ~-
+ x,
1 4
ul,,=~ - ~-~ + x + 1.
1 4 + x + y. We compare it The exact solution of this problem is u = ~x with the approximate one v = u + xysin(Trx)sin(Try*). To find the best upper bound for the error e, we minimize the majorant on the spaces Ys* constructed by means of the polynomial trial functions x i y j, it j = O, 1, 2 .... Fig. 6.10.9(a) shows the distribution of local errors and the errors computed by minimization of the majorant on Y* with 3, 6, and 10 trial functions are depicted in Figs. 6.10.9(b), 6.10 .10 (a) , and 6.10.10(b), respectively. The numbers are given in Table 6.10.3. We see that the DEM furnishes a good estimate of the energy norm of the error and of its shape even for N = 6. If N = 10, then the norm and the shape of the error are found with a high accuracy. Certainly, in more complicated examples it may be necessary to invoke more trial functions in order to accurately estimate an error and its distribution. Moreover, if gl has a complicated structure, then "global type" trial functions that we used may be inconvenient. In such a case, it is better to construct Y,* by local functions usually used in finite element methods.
6.10.5
Duality error proximations
majorants
for finite
element
ap-
Now, we focus our attention on finite element approximations of 2-D boundaryvalue problems. We consider problems with homogeneous boundary conditions in domains with polygonal boundaries. Results obtained by the DEM method are compared with other three error indicators widely used for finite element approximations. They are as follows: 9 the indicator ~72 - I I A V u h
- G(AVuh)ll 2
194
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
Figure 6.10.10" Integrands of the majorant for (a) N - 6 and (b) N - 10.
6.10. PRACTICAL IMPLEMENTATION
195
where G(AVuh) is a piecewise affine function defined at the central point xi of a patch 12i by the relation (cf. (6.4.6))
IT,ll
G(AVuh)[x,
(AVuh)[T~.
9 the indicator 72 based on the L2-projection of AVuh onto the space Vh of piecewise affine vector-valued functions (cf. 6.4.10): 72
-
inf
y* E Vh
2 IIAVuh-y*ll~.
9 the indicator _
IEI
IIj(aVuh" v)ll 2,
E6~h
where ~h is the set of interior edges and u is the unit vector normal to the edge E. The error indicators 71 and 72 were considered in Chapter 4. It is easy to see that 773 coincides with the part of the explicit residual error estimator that contains jumps. It is known (see, e.g., [46]) that for low order finite element approximations usually edge terms dominate in the residual type estimates and, therefore, can be used as an error indicator. We compare the results obtained by these indicators with those computed by D EM, paying attention to (a) the accuracy of the energy error estimate and (b) the quality of the local error indication. It should be noted that in many cases (especially for problems with smooth data), all indicators demonstrate close results. However, in other cases results are different. Below, we present several examples taken from [77], which show that various situations may occur.
In the first pair of examples, we consider two cases that differ only by coefficients of the matrix A. As we will see, the "standard" indicators give good results in the first case and are much less reliable in the second one. The error estimates computed by the DEM technology are quite reliable in both cases.
Example 6.10.1. Let us consider the equation div AVu + f = 0
C H A P T E R 6. LINEAR ELLIPTIC PROBLEMS
196
in a square domain f~ E ~2 and take f- 1 f = 0
forx E (0.5, 1.5), otherwise,
all = 10 all - 1
forx E (0.5, 1.5), otherwise,
a12 - a21 - 0 , a22 - 1. In this example, all the tests are computed for a certain fixed partition of the domain. To visualize the error indication, we mark elements by two colors (see Fig. 6.10.11): dark (if the error on a selected element is less than the mean value computed for the whole domain) and light (in the opposite case). This information can be further used in the simplest mesh-adaptation procedure that refines elements of the second group. The quantity p shows the number of correctly classified elements (in percent). The results of error estimation are compared with the actual values of errors obtained by using exact solutions. If in an example considered the exact solution is unknown, then the so-called "reference" solution is used instead. Such a solution is rigorously computed on a mesh that is much finer than those used in the error estimation tests. In all the examples, the mesh used for reference solutions has eight times more degrees of freedom than a primal mesh. Numerical results are collected in Table 6.10.4. In this example,
Ill
UI'-iii
-
ill- 0.02578
and the normalized value of the error is
ili eh ill = 5.3%. ill iii The respective results for the effectivity index are depicted in Fig. 6.10.12. From the values of p in Table 6.10.4 and the diagrams in Figs. 6.10.11 and
estimate I p
Table 6.10.4: r/1 ~2 0.03060 0.02952 1.19 1.15 90.2 90.3
r/3 93.0
v/M 0.02656 1.03 98.9
6.10.12, we conclude that the DEM gives an accurate error indication and a sharp upper bound of the error norm (p - 98.9% and I = 1.03). The
6.10. PRACTICAL IMPLEMENTATION
197
behavior of the effectivity index depicted in Fig. 6.10.12 shows that one time unit is enough for obtaining a sharp error estimate. It is easy to see that the majorant can also be used as an effective indicator of local errors. We present the true distribution of these errors and their estimates computed by the majorant as 3D graphs that present their averaged values on each element (see 6.10.12). By comparing these two pictures, we observe that the majorant accurately shows the distribution of local errors. The indicators ~1-~3 are also quite good. Among them, ~3 (P = 93.0%) is the best (but this indicator does not give a guaranteed upper bound of the total error). However, the quality of these estimates is lower than of those made by the D EM method.
Example 6.10.2. In the second example, we take the same data as in Example 1, but define the coefficient all as follows: all - 1 a~ 1 = 10
forx E (0.5, 1.5), otherwise.
In this case, the energy norm of the error is 0.02608. The respective results are collected in Table 6.10.5 and Figs. 6.10.13 and 6.10.14. It is clear that here, as in the previous example, the DEM technology provides a high accuracy of the error estimation. It was observed that, in the minimization
estimate I p
Table 6.10.5: r]l r/2 0.03346 0.03211 1.28 1.23 61.1 62.5
r]3 63.3
v/M 0.02680 1.03 97.9
process, the second term of the majorant diminishes with respect to the first one. As in the previous example, finding a sharp energy estimate takes about two time units. In contrast to the DEM, all indicators give an inaccurate error distribution (see Fig. 6.10.13), but ~1 and ~2 provide quite good estimates of the global norm. The above results show that, in some situations, the indicators r/l, ~2, and ~3 may work quite good, but in another one their quality may be not good enough to guarantee the most effective mesh-refinement. Since it is difficult to predict a priori which case we are dealing with, the quality of such error indication may be not optimal. The DEM provides quite reliable error estimates in all cases, but the determinition of a sharp distribution of local errors may require an extra CPU-time. We also remark that the
198
C H A P T E R 6. LINEAR ELLIPTIC PROBLEMS
Figure 6.10.11"
6.10. P R A C T I C A L IMPLEMENTATION
Figure 6.10.12:
199
200
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
Figure 6.10.13"
6.10. P R A C T I C A L IMPLEMENTATION
Figure 6.10.14:
201
202
C H A P T E R 6. L I N E A R E L L I P T I C P R O B L E M S
indicators give some values that are difficult to improve. In contrast, the quality of the DEM can be increased up to any required level (it is only the matter of time). In [77, 181], the reader can find a more detailed discussion of these questions and results of various numerical tests, in which the known a posteriori error estimation methods for finite element approximations were compared with those computed by the duality error majorants. At the end of this part, we make the following note. Since majorants contain constants depending on the respective domain fl, we must either find them analytically or compute their upper bounds by a certain numerical procedure. For problems with Dirichl~t boundary conditions such a procedure is easy to construct. In other cases, this task my require a more complicated analysis (see [145]). Finding such global constants is an important task that requires a deeper investigation.
6.10.6
Computational c o s t s
Let us estimate computational expenditures required to compute an upper error bound by means of the duality error majorant. Assume that f~ is a square domain and a finite element sampling is given by a regular (n • n) mesh with 6 elements in a patch. For large n we can estimate the total amount of elements N by the quantity 2n 2 and the number of edges M by 3n 2. In this case, the computation of the majorant (l+/3)l]Vu h
--
y*l12,• + ( 1 + ~ ) C ~ , , d i v y * + f , , 22 , f l
requires the following steps: (a) compute Vuh on each simplex T (which amounts to approximately N simple computations); (b) find the averaged gradient y* = G(Vuh) (the respective expenditures are proportional to the number of nodes and require only simple summation operations); (c) find y* - Vuh for any simplex T (approximately N simple computations); (d) find div Vuh for any simplex T (approximately N simple computations); (e) find one global constant C~. This step may not lead to extra computations provided that C~ (or an upper estimate of it) is defined
6.10. P R A C T I C A L I M P L E M E N T A T I O N
203
analytically. Otherwise, it leads to a single complicated computation of a constant. However, it is important that this constant does not depend on the mesh, so that in a process of mesh refinement no further computations of this constant are necessary; (f) compute N + N volume integrals and find M s (2N simple computations). Certainly, the upper bound of the error computed in such a simple way may be coarse. However, changing y* by some minimization procedure we can rapidly decrease the value of the index up to the numbers about 1.5 or 2. Further improvements of the effectivity index usually requires more time. We may compare these expenditures with the costs required to compute the explicit residual type estimate N
M
c2ijllj(vij 9 VUh)ll 22, E i j
cx,II div Vuh + fll~,T, + i=1
"
i,j=l (i>j)
In this case, we need to perform the following steps: (a ~) compute Vuh on each Ti (N simple computations); (b ~) compute normal vectors vii on each edge Eij (M simple computations, which, however, must be repeated upon changing a mesh); (c ~) find vii 9VUh for any simplex T (approximately 3N simple computations); (d ~) find div Vuh for each simplex T (N simple computations); (e ~) find the constants cli and c2ij ( N + M complicated computations, which must be repeated upon changing a mesh); (f) compute N volume integrals and M line integrals (N + M simple computations). It is not difficult to see that (a) and (a') are identical, while (b), (c), (b'), and (d) are not expensive and lead to comparable expenditures. The steps (d) and (d') are identical. However, (e) is much simpler than (e'), because we need to estimate only one global constant. In (d), we define local constants that depend on the mesh and must be recomputed after each mesh refinement. Even if such a recomputation is based upon some a priori computations made for a certain referenced element (or elements), the determination of all constants must take into account new geometric data
204
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
(angles, areas, etc) and would not be cheap. Also, a change in the degrees of polynomials forming the approximation space may lead to an expensive recomputation of all interpolation constants. Finally, we observe that (f) and (f) lead to approximately the same expenditures. Thus, we conclude that expenditures required for computing the majorant for a certain y* (e.g., for y* = G(AVuh)) does not exceed those that must be spend for the standard residual based error estimator. 6.10.7
Conclusion
Functional type a posteriori error estimates are universal error estimation tools that can adequately deal with conforming approximations of all types. Its combination with an effective numerical method for solving Problem P can give a reliable numerical strategy able to find approximate solutions with any a priori given accuracy. If an approximation v is close to the exact solution, then the DEM method often gives a good upper estimate even on the first step (e.g., without any minimization of M$ with respect to y*). In this case, we define y* by a simple averaging procedure and find an optimal parameter ~ in Mr y*, ~), which is a simple and easily solved task. In general, finding a sharp error estimate requires solving an additional minimization problem whose difficulty is comparable with the difficulty of the original problem. This effectively means that real expenditures for finding approximate solutions with a guaranteed accuracy can noticeably exceed those required for finding a solution without verifying its accuracy. We believe that, in general, such a "reliability fee" is unavoidable and one cannot obtain a posteriori error estimates that are simultaneously universal, exact, and computationally cheap.
6.11
Comments
At the end of this section, we prove one useful result that justifies the asymptotic behavior of the majorant Me usually observed in numerical experiments. P r o p o s i t i o n 6.11.1. Let (~k, Y~) E I~_ x Q* be a minimizing sequence for the majorant Mr ~, y*) and f~k ~ O.
(6.11.1)
Then the first term of the majorant tends to the energy norm o] the error and the second one to zero.
6.11. COMMENTS
205
Proof. Since (/3k, y~) is a minimizing sequence, we have M$(v,~k,Yk) - (1 + 3 k ) D ( A v , Yk)+
1 +-~k
-2 lie + A*y;ll 2
where C is a positive constant independent of k. bounded, so that for all k we have f~k _/3. Hence,
III Yk, [112.<- C
and
c,
The sequence f~k is
1 A* p, 2 ~k-kll (Y~ -- )il --< C.
(6.11.2)
From this fact, we conclude that there exists a subsequence (for the sake of simplicity we keep for it the same notation y~) such that y~ -~ y*
in Y*,
(6.11.3)
A'y; -+ A'p*
in U.
(6.11.4)
Therefore, lim M e (v, 3k, Y~) ___
m--+ r
> lim D(v,y~r > - m--,oo - 51 111Av III2 + ~1 [HY* 1112. -- (Y* , Av).
(6.11.5)
Recall that (3k, Y~) is a minimizing sequence for Me, so that ~lim _ ~ M e (v, f~k, y~) - ~1 OilA ( v - u)ill ~ and, therefore, 1
1 1 y, 2 ll1 h (v - u) Ill2 _> 2 111Av Oi2l + ~ IJi Oil. - (y*, hv).
This inequality leads to another one 1 p. _ 1 y* 2 - (p*, hv) + ~ [l[ 1112> - (Y*, hv) + ~ Ill Ill,.
(6.11.6)
Note that lim (y; - p*, Aw) + (p* - y*, Aw) - 0,
k--++c~
Vw e Vo.
In view of (6.11.4), the limit on the left-hand side is zero, and we find that (y*, Aw) - (p*, Aw),
Vw E Vo.
CHAPTER 6. LINEAR ELLIPTIC PROBLEMS
206
Take w = v - uo E Vo and use the relation (y* - p*, Av) - (y* - p*, Auo) + ( y* - p*, A(v - u0) ) - (y* - V*, Auo). Then, (6.11.6) yields 1
p,
(p*, Auo) - ~ [[1
_
1
[112,< (y,, Auo) - ~ 1[[y 9 1112,.
(6.11.7)
Since p* is a maximizer of Problem ?*, we have
I*(p*) >_I*(y*) which is equivalent to 1
p,
(p*, Auo) - ~ 111
2
1
11[,_> (y*, Auo) - ~ [[[
y,
[112,
.
(6.11.8)
Now (6.11.7) and (6.11.8) leads to the equality 1
p,
(p*, Av) - ~ 111
2 1 y, 2 [11,- (y*, h v ) - ~ Ill 11[,.
(6.11.9)
However, Problem P*, has only one maximizer, so that (6.11.9) means that y* = p*. Now, we obtain 1
Ill A ( v - u)1112- m-~oolimMr
#1 + #2,
(6.11.10)
where #1 #2
-
-
lim (1 + ~k)D(Av, y;),
k--+oo
lim
1+
2- Ill + A*p~ll ->" 0
We know that y~ weakly converges to p* in Y*. Therefore (we use lower semicontinuity of the norm), 1
l im (~ I[] A v ill2 + ~ []] y~ ]]]2,-(AV, Yk))>_
k--+oo
1 1 p, 2 1 - 5 [li hv Ill2 + ~ [1[ lU, - (hv, p*) - ~ ][I h (v - u) Ill2
6.12. NOTES FOR THE CHAPTER
207
and from (6.11.10) we conclude that #1 - ~1 ][[ A(v-u)I[[ 2 and #2 - 0. Now, we have 1 1 p. 2 , ) #~ - ~ I[I Av I[[2 + ~ [[[ [[[, - (Av p* =
. + Z~ D (Av, y~) .) -~oo 2 ]Uhv [[[2 + 51 [[[ yk. [][2._ (Av, yk)
lim ( 1
=
1
-- -2 j[[ Av I[[2 - ( h v ' p*) + kl im 51 IIIYk, 1112,9
Thus, ][[ y~ U[.-~lll p* ][Ik and, consequently, y~ strongly convergence to p*. In view of (6.11.4), we conclude that y~ strongly converges to p* in Q*. [:]
Remark 6.11.1. Such a behavior of the majorant (when the first term tends to the energy norm of the error and the second one rapidly decreases) was really observed in the vast majority of numerical tests. Some of them are presented in [77, 78, 173, 181]. Remark 6.11.2. Consider linear elliptic equations of the second order. In this case, Q* - H(div, f~). Assume that y~ converges to p* in Q*, then it is not difficult to show that the integrand of M e (denote it by #(x)) approximates the integrand of the error (denote it by e(x)) in the following sense. Let e > 0 be an arbitrary small number and
It is proved that [w[ tends to zero as k tends to infinity (see [182]). This fact justifies local properties of the majorant that can also be used as an efficient indicator of local errors. 6.12
Notes
for
the
Chapter
The history of the variational approach to deriving a posteriori error estimates dates back to the works of W. Prager and J. L. Synge [163] and S. Mikhlin [143]. In [143], it was shown that if a problem is stated as a minimization problem for a quadratic functional J, then the difference between J(v) and the exact lower bound provides an upper bound of the error. Since the latter is bounded from below by the values of the so-called dual (or complementary) functional, it is natural to use the respective dual problem for the computation of a posteriori error estimates (see, e.g., H. Gaevski~, n. K. Grhger and K. Zacharias [82], D. W. Kelly [118] M. Mosolov and V. Myasnikov [146]). Such bounds can also be computed by the orthogonal projection method (see S. Zaremba [221], H. Weil [218], and M. I. Vishik [215]),
208
C H A P T E R 6. L I N E A R ELLIPTIC P R O B L E M S
which creates a sequence of trial functions for the dual problem. However, in the majority of cases admissible functions of the dual problem belong to a linear manifold defined by some differential relations (we denote it Q~). In general, the determination of such functions poses a difficult task. Probably, this fact hindered the development of a posteriori error estimation methods based upon duality theory. For a class of variational problems with convex integrands dependent on the gradient of the minimizing function, a way of overcoming this difficulty was found and justified in S. Repin [168, 170, 171]. In S. Repin [174] and some other papers, such a justification is presented for a wider class of uniformly convex functionals. Functional type a posteriori error estimates for boundary-value problems with the biharmonic operator were derived in P. Neittaanm/iki and S. Repin [149]. Two-sided a posteriori estimates of the functional type were derived in S. Repin [175] for linear and some nonlinear elliptic problems. The variational approach described in Chapter 6 is based on these results. Also, [175] presents another (nonvariational) method of deriving a posteriori estimates. With the help of this method, estimates of the deviation from the exact solution were obtained for a problems that have no variational statement (see [176]). For linear elliptic equations of the second order, a numerical verification of a posteriori error estimates of the functional type was carried out in the papers by S. Repin [173], M. Frolov, P. Neittaanm~iki and S. Repin [77, 78, 181]. For equations of the fourth order, several tests were performed in M. Frolov, P. Neittaanm~iki and S. Repin [78]. For linear elasticity problems, such tests can be found in A. Muzalevskii and S. Repin [147]. Finally, we mention several results related to the subject considered, which has not been exposed in Chapter 6. Functional type a posteriori error estimates for approximations that do not satisfy Dirichl6t boundary conditions precisely were obtained in S. Repin, S. Sauter and A. Smolianski [182, 183]. Such estimates can be useful if it is difficult to precisely approximate the boundary or if a numerical method takes into account boundary conditions in a generalized sense (see, e.g., R. Glowinski, T.-W. Pan and J. Periaux [92]). In S. Repin [l $0] and S. Repin, S. Sauter and A. Smolianski [184], functional type a posteriori error estimates were used for estimation of dimensional reduction errors arising when a 3D model is approximated by a simplified 2D model.