Evolution Equations and FEM

Chapter 3 Evolution Equations and FEM

In the case of temporally inhomogeneous parabolic equations, the complex m e t h o d does not work so well by itself for the analysis of the finite element approximation. Here, we adopt the m e t h o d of Helfrich and then that of energy, and extend the error estimates of the preceding chapter to this case.

3.1

Generation

Theories

The present chapter is devoted to the temporally inhomogeneous parabolic equation; t2 C R 2 denotes a polygon, and /2 = /2(x,t, D) a second order elliptic operator with time-dependent real smooth coefficients:

c(~, t, D) = - ~

~-~ a~j0 (~, t)

0

2

+ Z~=~ bj(x , t) ~+~(~,t). o

i,j=l

Uniform ellipticity 2

i,j=l

is assumed, 51 > 0 being a constant. We study the parabolic equation

ut +/2(x, t, D)u = 0

in t2 x (0, T)

(3.1)

with the initial condition

glt=o = uo(x)

in ft

(3.2)

and with the boundary condition either u = 0

on at2 x (0, T)

(3.3)

or

0u -t- a u = 0 cO~,c

on 0t2 x (0, T ) .

--

(3.4)

In (3.4), cr = a(x, t) is a smooth function on 0t2 x [0, T], and 0/0r, L denotes the differentiation along the outer co-normal vector ~L: 0

_ ~-'2

n i a i j ( x , t)

i,j=l

95

0

3. Evolution Equations and FEM

96

where n = (nl, n~) is the outer unit normal vector on 0f~. Assuming u0 E X = L2(~), we can reduce equation (3.1) with (3.2) and (3.3) (or (3.4)) to the evolution equation

du

dt

+ A(t)u = 0

(0 < t < T)

(3.5)

with u(0) = u0

(3.6)

in X = L2(gt). Taking V = H~(f~) or Hl(f~) according as the b o u n d a r y condition (3.3) or (3.4), we put

.At(u, v) = i,j:,

aij(x, t)-~xj -~xi dx + J:'

bj(x, t)--VOzy dx

+ ffl c(x, t)uv dx + fon a(x, t)uv dS for u, v E V. An m-sectorial operator A(t) in X can be defined through the relation

At(u, v) = (A(t)u, v) ,

(3.7)

where u E D (A(t)) C V and v E V. As in the temporally homogeneous case of w relation

D (A(t)) = H2(fl) N H~(fi)

the

(3.8)

follows for V = H~(f~) and

D (A(t)) = {v C H2(f~)

+av=O

on0f~

(3.9)

for V = Hl(f~), if 0fi is smooth for instance. Generation theories of a family of evolution operators {U(t,S)}o
du dt

- - + A(t)u : 0

(s < t < T)

(3.10)

with 'u(s) = 'Uo C X

(3.11)

3.1. Generation Theories

97

is given by u(t) = U(t, s)u0. Consequently, the semigroup properties

U(t,r)U(r, s) = U(t, s)

(0 < s < r <_ t <_ T)

and

U(t,t) = I

(0 <_ t <_ T)

follow. Furthermore, we have

U(t, s) = - A ( t ) U ( t , s)

and

-~sU(t, s) = U(t, s)A(s)

(3.12)

in the strong sense in X for 0 _< s < t < T. It is worthwhile to give a short summary of those theories here. In fact, studying the finite element approximation of (3.5), we show some kind of stability of approximate solutions and smoothness of original ones. Both of them are established by re-examining the theories, and then error estimates will follow. Crucial assumptions are the following: (i) Each - A ( t ) generates a holomorphic semigroup with certain estimates uniform in t E [0, T]. (ii) A(t) is smooth in t E [0, T] in some sense. The family {U(t, S)}o
IA,(~, ~)1 s c~ II~llv II~llv

(u, v ~ V)

A~(~ , ~) > a l~ I~V - ~1~1 X=

(~ ~ v)

and -

-

,

(3 13) "

with constants C1 > 0, 6 > 0 and A E ][{ independent of t E [0, T]. We may suppose A = 0 in (3.13), taking v - e-~tu(t)instead of u - u ( t ) i n (3.5)"

x,(~, ~)>__ ~11~11~

(~ ~ v ) .

(3.14)

Then, A(t) becomes of type (0, M) with 0 E (0, 7r/2) uniformly in t e [0, T]; we have

C \ Eo C p(A(t)) for

ro = {~ c e l

o < arg I~l < o},

(3.15)

3. Evolution Equations and FEM

98 and M~ II(zI- m(t)l _ XlIx, X <_-~

(0 + e _< [arg z I < _ 7r)

(3.16)

for e > 0. Each -A(t) generates a holomorphic semigroup {e -~a(t) }s>0 uniformly bounded in t C [0, T]. We have

II~-~A(,)llx,x <_1 for 0 < s < + o c and 0 < t < T. Concrete expressions of the second condition, on the other hand, are slightly different according to the theories. In each of them, construction of the family gives some estimates on their members at the same time. For the moment, the operator norm on X = L2(12) is simply written as II" IIGeneration

T h e o r y of T a n a b e - S o b o l e v s k i i

Because of (3.8), D(A(t)) is independent of t in the case of V = H~(f~). Furthermore, the inequality I[A(t)A(s)

-~ -

111_

cIt-

sl ~

(t,~ E

[0, Z])

(3.17)

follows with c~ C (0, 1] from an integration by parts and the elliptic estimate. In such a situation, the family of evolution operators {U(t,s)}o
U(t,s) = e-(t-s)A(~) +

e-(t-r)A(~)R(r,s) dr,

(3.18)

where R = R(t, s) is the solution of the integral equation of Volterra type, I~,(t, S) --

I~ 1 (t, r ) R ( r ,

s)dr

= /~1 (t, S)

(3.19)

for /~,1 (t, S) -- --

(A(t)

- A ( s ) ) e -(t-s)A(s)

Those relations are obtained formally by substituting tile right-hand side of (3.18) into the first relation of (3.12). Eventually the unique solvability of (3.19) follows and {U(t,s)} defined by (3.18) becomes the desired family of evolution operators. Meanwhile the following estimates are proven, where 0 _< s < r < t _< T and 0 E (0, a):

(3.20) IIU(t, s)ll + [IA(t)u(t, s)A(s)-' I[ <- c,

[IA(t) [ u ( t ,

~) - u(,-,

~)] A(.~)-' II -< C o ( t

- ~)o(~ _ ~ ) - o

(3.21) (3.22)

3.1. Gel~el"ation Theories

99

G e n e r a t i o n T h e o r y of F u j i e - T a n a b e If V = Hl(f~), D(A(t)) varies as t changes, and the preceding theory does not work directly. However, the m-sectorial operator A(t) in X defined through (3.7) can be regarded as that in V', which is denoted by A(t). Its domain is independent of t: D(A(t)) = V, and furthermore, conditions (3.15) and (3.16) hold with A(t) and X replaced by A(t) and f( = V', respectively:

V',V' --

]Z q- 1

The coefficients aij, bj, c and cr are smooth so that the inequality IAt(u, v) - A~(u, v)l _< O It - sl ~

I1~11~I1~11~

(u, v E

V)

(3.24)

holds with c~ E (0, 1]. This implies (3.17) for A(t) in 2 . Therefore, from the preceding theory, {A(t)} generates a family of evolution operators in 2 denoted by {U(t, s)}0 1/2, it can be shown that

U(t,s)=O(t,s)x becomes a bounded operator in X and {U(t,s)} forms a family of evolution operators generated by - A ( t ) . This {U(t, s)} satisfies (3.20) and U(t, s)[ _< C in X. G e n e r a t i o n T h e o r y of K a t o - S o b o l e v s k i i We can show that D (A(t) p) is invariant in t for p C [0, 1/2). symmetric part A ~ of At as in w

To see this, take the

Ato= 1 (.,4, + At).

At(~, ~) = A,(~, ~),

If Ao(t) denotes the self-adjoint operator associated with At~ it holds that

c - ' Ivllv _< IlAo(t)l/% Ix < C vllv.

(3.25)

As is described in the commentary to w the domain of A(t) p is equal to that of Ao(t) p if p C [0, 1/2). (The idea of proof will be obtained from the argument described below.) Here, the latter is independent of t if p = 1/2 as is indicated in (3.25), and p = 0 obviously:

D(Ao(t) '/2) = V,

D(Ao(t) ~ = X.

Then, Heinz' inequality implies the invariance of D (A(t) p) in t for p E [0, 1/2). We show that the inequality

IlA(t)PA(s)-p-/11

c t - sl

(t, s E [0, T])

(3.26)

3. Evolution Equations and F E M

100 follows from (3.24). In fact, given # > 0, we have (#I + A(t)) - 1 -

(pI + A ( s ) ) - ' = - (pI + A(t)) -1 Ao(t) 1/2. Bo(t, s ) . Ao(s) 1/2 (#I + A(s)) -1

with Bo(t, s) = Ao(t) -1/2 [A(t) - A(s)] Ao(s) -~/2

Inequality (3.24) reads; IA(t) - A(s)l v,v, <- C I t - sl ~ , or

liB(t, ~)11 ~ c It - ~1~ We make use of the following. L e m m a 3.1. If H is a non-negative self-adjoint operator on a Hilbert space X with norm

II. II, ~ e h~ve

~

~ #2~ IH~/2 (~I + H) -~ v[I 2 d # -

Cp IIH~

2

(3.27)

for p c [0, 1/2) and v E D(H~ Proof: Introducing tile spectral decomposition

ad E ( a) ,

H :

we have

#~o ]IH~/~ (#I + H)-'

= d#

-

~,=o @

a(~, + a ) - ~ d IIE(a)~ll ~

9

-

ad IIE(:~)vll =

=

ad IIE(:~)vll ~. a ~o-'

=

Co

#~o(# + a)-~da ~o(#

+ 1)-2d#

A2PdlE(A)vl 2-- CplIHPv 2.

This means (3.27).

[]

We are able to give the following. Proof of (3.26): We show

II[A(t) p - A(s) p] vii < c It - sl ~ IIAo(s)~

(3.28)

3.1. Generation Theories

I01

for p E (0, 1/2) and v e D (Ao(s)a). Then, inequality (3.26) follows from (2.43). For this purpose, we may take v r D(A). Applying (2.18), we have ([A(t) ~ - A(s) ~ v, X) = sin 7rp

7r

/0

pP dp. (B(t,s)Ao(s) 1/2 (pI + A0(s)) -1 v, Ao(t) 1/2 (pI + Ao(t))-i X)

for X E X. In use of (3.25), inequality (3.28) follows as [([A(t)P - A(s)P] v, X)[

(/0 9

I[Ao(t)'/2 (#I + Ao(t)) -x Xll2x d•

It - s[ ~

= c~ IIAo(s)Ovllx. I l x l l x - I t - sl ~ . The proof is complete. Above properties of the fractional powers of generators, summarized as the invaria.nce of D (A(t) o) and inequality (3.26), guarantees the existence of the family of evolution operators in the following way if c~ + p > 1 is satisfied. Therefore, evolution operators exist if c~ > 1/2 again, in the parabolic initial b o u n d a r y value problem in consideration. For simplicity, we describe the case p = 1/m with m an integer. By taking an appropriate approximation of A(t), say, the modified Yosida approximation: Aa(t) = A(t)(1 + AA(t)o) -m with ,~ I 0, we can reduce the theory to the case where each A(t) is bounded in X. In fact, we can show that (3.26) implies

IlA~(OPA~(s) - p - Ill <_ C l t - sl ~

(t, s e [0, T])

with a constant C > 0 independent of )~ > 0. Existence of a family of evolution operators {Ua(t,s)}o
U(t, s) = s-lim Ua (t, s). M0

Then, {U(t,s)}o
llA(t)ZU(t,s)ll < C z ( t - s)-z

(O<_s
follows from ]lA~(t)Zu~(t, s)[I < C z ( t -

s) -z

(0 < s < t < T ) ,

(3.29)

where 0 _< p < a + p. We shall give an outline of its proof, dropping the suffix A for simplicity of writing.

3. Evolution Equations and FEM

102 Letting

D(t, s ) = A(t)PA(s) - p - I, we have

liD(t, s)

~

Clt

-

s ~

and m

A(t) - A(s) = Z

A(t)I-PPD(t' s)A(s)PP"

(3.30)

p=l

We get t

-

= ~=

e-(t-r)A(t)A(t)l-p"D(t,r)A(r)~~

dr.

(3.31)

We introduce the following notation: Given the operator-valued functions I(e = IQ(t, s) (g = 1, 2) on D = {(t, s ) [ 0 _< s _< t <_ r } , we define another I( = K1 * I(2 by (//(1 * 1(2)(t,s)=

K~(t,r)K2(r,s) dr.

For a > 0 and M > 0, we say t h a t K E Q(a, M) if the inequality

[ K( t,s)ll - M ( t -

s) a-1

holds. Then, if Kl C Q(ae, Me) for g = 1, 2 we have /s

* /(2 E Q ( a l +a2, B ( a l , a 2 ) M 1 M 2 ) ,

where B(a, b) denotes the beta function: B(a,b)

jfo 1(1

=

-

x )a- l x b-1 d ~ .

Let

W(t, s) = U(t, s) - e -(t-~)A(t)

and

Yq(t, s) = A(t)oow(t, s).

Equality (3.31) reads; m

(3.32) p=l

3.1. Generation Theories

103

where

Hq,p(t, s) = A(t)'-PP+qPe-(t-s)A(t)D(t,

s)

(3.33)

and m

p=l

for

Yp,_, (t, 8) = A(t)PPe-(t-s)A(t) To avoid a technical difficulty, we take (3.34) and transform (3.32) into a system of integral equations for

{Zql q = 1,.-. ,m}. That is, m

Z~ = ~

Hq,, , Z, + &,o

(3.35)

p=l

with m

(3.36) p=l

Consequently, Zq (q = 1 , . . . , m) is subject to the iteration scheme oo

Zq -- ~

(3.37)

Zq,i

i=0

with m

Zq,,+, = ~

H~,, 9&,,

(~ = o, 1 , . . . ) .

p=l

From definition (3.33) follows H~,, ~ Q (~ - qp + pp, M , )

with a constant M1 > 0, because (2.16) holds by (3.16)"

<

> 0),

(3.38)

3. Evolution Equations and F E M

104 where ~c >_ 0. F u r t h e r m o r e , we have

Zq,o E Q(1 + a - qp, Mo) for some M0 > 0. We can derive

Zq,i E Q (1 + (i + 1)a - qp, Mi) with A/Ii+l/~Ii = r n M o M i B (c~ + p - 1, (i + 1)c~) by an induction, and then Zq E Q ( 1 + a - q p ,

(3.39)

C)

follows from (3.37) with a constant C > 0. Inequality (3.39) now gives an estimate on Yq even in the case t h a t q is not an integer. Namely, (3.34) makes sense for any q >_ 0, and Zq,o is again given by (3.36). T h e n Zq is defined by (3.35) with p taking integers as before. Since Zq in (3.35) admits (3.39), inequality (3.34) gives an estimate of Yq for a - qp + p > 0, because Yq,O E Q(1 - qp, C) can be shown. Consequently, Yq E Q(1 - qp, C) follows. Writing qp = / 3 , we then obtain (3.29) from (3.38).

Generation Theory of Kato-Tanabe Let us define another bilinear form .At(, ) on V x V by

,=

j=l

s

/o ~

Then, we have

IA,(u,~)l _< C llullvllvllv, I.A,(.,,, v) - A,(.,,, ~)1 _< c It - ~1~ II~llv Ilvll~, for u , v E V with some c~ E (0, 1] and lira

sup

(At-

1

As)('u, v) - As(u, v) = 0 .

II~llv,llvllv
dA(t)-' dt

d A -

ds

(S)-1 ~

holds with K > 0 and a E (0, 1].

K

It-

s

(t, ~ E [0, T])

(3.40)

3.1. Generation Theories

105

2) The inequality

c9 ( z I - A(t)) -~

M~

holds for c > 0 sufficiently small.

Proof of (3.41): Take z

(z e c \ r0+~)

---[;i

9 C \ E0+E and v

(3.41)

9X, and put

w(t)=(zI-A(t))-lv. We show that it is continuously differentiable in X with the estimate

Ow x

-5i-

M~

(3.42)

-
satisfied. Recall

(8.48)

(z - A ~ ) ( ~ ( t ) , ~) = (~, ~) and (2.31)"

Jzl lJxJl~ + II~II~V - -<

~o I~ II~IIX~

- A ( ~ , ~)J ,

where X C V and z C C \ Eel. This implies also (2.33) as

I1~1 ~ < 6o Iv ~ /I~1

~nd

I1~11~ < ~o vlt~ /t~ ~/~

Differentiating formally in t, we get (Z -- , A t ) ( ~ b ( t ) , )(~) = A t ( w ( t ) , ) ~ )

(3.44)

from (3.43). Here, the right-hand side is a bounded linear functional of X C V. In use of (2.31), we can define ~b(t) E V through (3.44) conversely, by Lax-Milgram's theorem. Now we shall show lim w(t) - w(s) _ ~b(s)ll t--~s

t -- 8

: 0.

(3.45)

V

In fact, letting ~ ( t ) - ~(~) t--8

we have

(~ -

A~)(~(t), ~(t))

=

t-8

t--8

(Z -- r

-- W(8), x ( t ) ) -- (Z -- ,At) (~b(8), x ( t ) )

(z - A~) ( ~ ( t ) - ~(~), ~(t)) - As (~(~), ~(t)) + (A~ - A~)(w(~), ~(t))

t--8

{1

((z - As) - (z - A~)) (~(~), ~(t)) - A~ (w(~), ~(t)) + (At - A s ) ( ~ ( ~ ) , ~(t))

}

3. Evolution Equations and FEM

106 Again by (2.31), we get 1

IIx(t)ll* V - -< ~0

(.4, - .4~) (~(~), z(t)) - A~ (~(~), ~(t)) + &c It- ~1. Iw(~) ,,. IIx(t) v,

or

IIx(t)llv < C I t - s

9~,(~)1 v +C

sup xEV

1

---zU t (.4, - .4~) (~(~), ~) - A~ (~(~), ~)

Ilxllv_
lim IIx(t)llv = 0 . t---* 8 We turn to the proof of (3.42). In fact, again by (2.31) we have

z IIw(t)ll x2 + I~(t) ~. _< ~o I(z - .A,.)(~(t), ~(t))l 5o 12., (w(t), w(t)) I _< C IIw(t) ,, IIw(t)ll,,. This implies

~i,(t) Iv < c IIw(t)Iv -

CIl~ll~ Izll/~

and

~/2 II~i~(t)lv/~ ......

IIw(t) Ix < Cllw(t)llv

Izl '/~


-

C lvllx

Izl

by (2.33). The proof is complete.

Proof of (3.~0): We take v C X, and set .f(t) = A ( t ) - l v E V. The existence of df(t)/dt follows similarly. Noting At (.f (t), X) = (v, X) , we define j~(t) by

where ~ E V. Then it follows that lim

t--~8

.f(t) - . f ( s ) - r t-s

=0. V

3.1. Generation Theories

107

Inequality (3.40) is a consequence of the stronger one:

Ilvll~,.

(3.46)

In fact we have

IIx ~ < A<(r - r x) = -Jlt(f(t), x ) - ( A t - A,)(j'(s), x ) + Jl,(f(s), x) = ( f % - Jlt)(f(t), x ) + J l ~ ( f ( s ) - f(t), x) - ( A t - As)(r <

x)

C Ilxllv (it - sl ~ IIf(t)llv + Ilf(s) - f(t)llv + It - sl IIr

or

iii(t)- ./(~)llv _< c (it- ~l ~ lif(,)il~ + ilf(~)- f(t)ll~ + it- ~l

Its( )ll ).

(3.47)

Remember that

IIf(t)llv <_ C IIv v, follows from

llf(t)ll~

_ .At(f(t), f(t)) -- (v, f(t)) < c

IlVllw, IIf(t)llw

.

Similarly,

~llJ(s)ll~ ~ A,(r

r

- -J~(f(s), r

< c }lf(s)llv IIf(s)llv

gives

Finally, we have

(5lif(s)- f(t)ll~

< .At(f(s)- f(t), f ( s ) - f(t)) = ( A t - As)(f(s), f ( s ) - f(t)) <

C It - ~1 IIf(~)ll. IIf(~) - f(t)ll.

and hence

] i f ( s ) - f(t)iiv <_ C I t - s I tlf(s)llv <_ C i t - s I livllv, follows. Plugging those inequalities into (3.47), we obtain (3.46). Subject to the above conditions on the differentiability of resolvents, the family of evolution operators {g(t,S)}o
u(t, ~) = ~-(~-~)~(~) + w ( t , ~)

3. Evolution Equations and FEM

108 with

w(t, ~) =

~-(*-~(*)R(~, ~) d~,

(8.48)

where R = R(t, s) is the solution of

R(t, ~) = R~(t, ~) +

R~(t, T)~(,-, ~) d~

(3.49)

for

0 + N0 R l ( t , s ) = -(~-t

)r

1 O~F e -(t- s)z -07(zI0 -- 27r,. A(t))-ldz,

(3.50)

F being positively oriented boundary of E0+~ for e > 0. Inequality (3.20) and IIg(t, s)ll < c are also derived from this scheme. More precisely, we have

IR(t, s)][ _< C

(0 _< s _< t _< T)

(3.51)

and

(0 _< s < ~ < t <_ T),

(3.52)

for 7 C (0, 1) and 5 C (0, a). They are consequences of those with R replaced by R1,

IIR~(t, s)ll <_ c

(3.5a)

and

II/~l(t, 8 ) - /l~l(r, 8)11 ~ C7,5((t- T)7(T- 8) -7 -[- ( t - r ) a ( r - s)~-a-1), respectively. This theory of generation is particularly remarkable as any assumptions on the domains of A(t) are not made. Consequently, it is no wonder that a little stronger assumption on the smoothness in t of A(t) is imposed.

3.2

A Priori

Estimates

In this section, we show a uniform estimate concerning the evolution operator associated with the discretized problem, which plays a key role in the error analysis. In the following lemma, V C H C V' denotes a triple of Hilbert spaces, A(t) an m-sectorial operator in X associated with the bilinear form At(, ) on V x V, smooth in t E [0, T] in the sense of the preceding section, and {U(t, S)}o<8
3.2. A Priori Estimates

109

L e m m a 3.2. Each/3 e (0, 1/2) admits the equality

d(t)U(t, s) - A(r)U(r, s) = A(t) [e -(t-~)A(t) - e -(~-s)A(t)] + A(t)ZZ~(t, r, s)

(3.54)

with Zz(t, r, s) subject to the estimate IIZ,(t, r, s)ll <

C,(t

-

r)(r

-

(3.55)

s) ~-',

where O < s < r < t < T. Proof: We shall take the reduction process first. In fact, from the construction we have A(t)U(t, s) - A(r)U(r, s) = A(t) [e - ( t - s ) A ( t ) --

e -(r-s)A(t)]

+A(t)" [A(t)l-'~ -(~-s)~(') - A(~)'-'~ -(~-s)~(~)] +A(t)" [ I - A(t)-ZA(r)'] A ( r ) l - ' e -(~-s)A(~) +A(t) a [ A ( t ) l - ' w ( t , s) - A ( r ) l - ' W ( r , s)] +A(t) ~ [ I - A(t)-~A(r) ~] A(r)'-~W(r, s), so that 4

/=1

holds with

Z~(t, r, s) = A(t)l-Ze -(r-s)A(t) - A(r)l-Ze -(r-s)A(~), Z~(t, r, s) =

[I - A(t)-ZA(r) ~] A(r)X-'e -(~-s)A(r),

z~(t, ~., s)

A(O'-~w(t,

:

Z~(t, r, s) :

s) - A(~.)~-~WO ., ~),

[I - A(t)-ZA(r) ~] A(r)'-~W(r, s).

The Dunford integral gives the expression

d(t)Ze_Tm(t ) _ A(s)e_~d(~ ) = 12m ofr z'-Ze-rz

[ ( z I - d ( t ) ) - ' - (zI - m(s))-'] dz,

while

]l(z~-

A ( t ) ) -1 - ( z I -

A(s))-'ll < c~ It- sI -

Izl

follows from (3.41). We have

IlA(t)~e -~A(t) - A(s)~-~(s)ll <_ c~ It - sl r - ~ < c

d--~-~lt-s l -

~ - ~ e -~"~~176 At

G I t - sl r ~-~

(3.56)

3. Evolution Equations and FEM

110

for/3 > -1. This implies IIz~(t, ~, s)ll < c~(t - ~)(~ - ~)~-~

Inequality

III-

A(t)-ZA(s);311 <- C ~ l t - sl

follows from (3.26). We have

~

C / 3 ( t - r ) ( l " - 8) ~-1

Inequality (3.51) gives IIA(t)l-ZW(t,s)lt

: <_ c~

A(t)l-ze-(t-r)A(t)R(r,s)dr ( t - ~)~-'d~ = C ~ ( t -

~)~

and hence llz~( t, ~, ~)11 - c , ( t - ~)(~ - ~),

follows. The proof of Lemma 3.2 has been reduced to

Ilz~(t, ~, s)l t = IlA(t)l-~w(t, s) - A ( r ) ' - ~ W ( r , s)l I <_ C,(t - r ) ( r - s ) ' - ' Equality (3.48) now gives 7

Z~(t,r,s) = A ( t ) l - Z w ( t , s ) - A ( r ) l - Z W ( r , s ) = ~

Z[3(t,r,s )

l=5

with Z~(t, r, s)

=

f

t

A(t)l-~e -(t-~)A(t) [/~(z, s) -/~(t, s)] dz,

z~(t, ~, ~) =

[ A ( t ) ' - ~ -('-z~('~ - A ( ~ ) l - ~ - ( ~ - ~ ( ~ ]

Z~(t, ~, s)

A(t)~-~-(~-z~(~)dz

=

+

[P(~, 9 .~) - ~(~, ~)1 d~,

. ~(t, s),

[ A ( t ) l - ~ - ( ~-z~(~ - A ( ~ ) I - ~ -(~-z~(~]

d z - ~ ( ~ , s).

3.2. A Priori Estimates

111

In use of (3.52) we have

[IZ~(t,r,s)l[

<

IlA(t)'-Ze-(t-z)A(t)ll .

<_ c~.,

IIR(z,s)- R(t,s)ll dz

( t - ~)~-~(t- ~)~(~- ~)-~d~

+G,~

( t - z)~-~+~(z- s) -~+~-~ (9z - s)~-'dz,

<- G .

( t - z)~-l+~(z- ~)-~+l-~dz. ( ~ - s) ~-~

=

~)(~- ~)~-1

G(t-

taking "7 e ( 1 - f l , 1) and 5 e (c~ - fl, c~). From (3.96) and (3.56)we have

_ G(t-

~)(~- z) ~-~.

Combining this with

}lA(t),-z~-I,-zl~r

A(~,)'-Z~-Ir-z/Ar II

-< G(~"- ~)~-', we get [IA(t)~-ze

-r

-

A(r)l-Ze-(r-z)m(r)[I < C ~ ( t - r)'~(r- z) - ~ + z - '

for r~ E [0, 1]. This implies

IIG( t, ~, ~)ll _<

I[A(t) ~-~-(~-~(~ - A(~)'-~-(~-z)~(~)l[- I1~(~, ~ ) - ~(z, ~)11 d~-

+c~,~

= G(t-

( t - ~)~(~ - ~)-~+~-1. (~ - ~)~(~- ~)~

,~)~(,~- s)~-',

3. Evolution Equations and FEM

112 by taking -7 e ( a - / 3 , 1) and (5 E ( a - / 3 , a) similarly. We have Z~(t, r, $)

=

[A(t)-fle-(t-z)A(t)]~:t r R(t, 9 8) -Jr-[A(t)-fle -(t-z)A(t) - A ( r ) - f l e - ( r - z ) A ( r ) ] : : :

I~(t, 9 8)

11

/=8

with

Z~(t, r, s) = [I - e -(t-~)m(t)] [A(t)-zR(t, s) - A(r)-ZR(r, s)], Z~(t, r, s) = - e -(t-r)A(t) [A(r) -z - A(t) -z] R(r, s), zl~~ r, s) = - [A(t)-Ze -(t-~)A(t) - A(t)-f~e -(~-s)a(t)] R(r, s), z ~ l ( t , 1", S)

:

-- [A(t)-~e -(r-s)A(t) - A(r)-~e -(T-s)A(r)] R(r, s).

The inequality

IIA(t) - ~ - A(~)-~II <- G It- ~1 follows from (3.26) so that we have IIz~(t,

~, ~)11 <-

c~(t -

~)(~ - s)/2i'- 1 .

On the other hand the inequalities

Ilzy(t, r, ~)ll _< Cz(t- ,~)(,~- ~)~-' and

IIz~'(t, r, s)ll _< G ( t - r)(~- s)~-' follow from (3.96) and (3.56), respectively. In this way, Lemma 3.2 has been reduced to

[[A(t)-zR(t, s ) - A(r)-ZR(r, s)[ I _< Cf3(t- r ) ( r - s) z-I

(3.57)

Integral equation (3.49) reduces this inequality furthermore to IIA(t)-ZR,(t, s ) - A(r)-ZR,(,', s)l [ <_ C ~ ( t - r ) ( r - s) z-l,

(3.58)

R~ - Rl(t, s) being the right-trend side of (3.50). We show this fact first. In fact, in use of (3.49) we have m(t)-~R(t,

s) - d ( r ) - r R ( r ,

s)

= [ A ( t ) - O R l ( t , s ) - A(r)-aR~(r,s)] + +

/'

A(t)-;3R,(t,z)R(z,s)

dz

[ A ( t ) - O R l ( t , z ) - A(r)-aR~(r,z)] R(z,s) dz.

3.2. A Priori Estimates

113

The second term of the right-hand side is estimated by (3.51) and (3.53):

/*

liA(t)-Zll

9IIRl(t,

z) l-IIR(z,

~)11 dz <

c(t

-

~).

Similarly, the third term is estimated as

fs r IlA(t)-ZRl(t,z) - A(r)-~t~l (r, z) C

dz

9IR(z,s)

( t - r ) a ( r - z)~-lds = C~3(t- r ) a ( r - s) l~.

Thus, (3.58) implies (3.57). We now turn to the proof of (3.58). Recall the symmetric part Ao(t) of A(t) associated with Ao. We have the following. L e m m a 3.3. The equality

[ 0_~( z l _ A(t))_ 1 - -~sO(zl - A(s))-l] = A(t)lo/2 ( z I - A0(t)) -1Bz(t, s)Ao(s) 1/2 ( z I - Ao(s)) -1 holds with

Bz(t, s)II ~

c I~ -

(3.59)

sl ~ ,

where z E EOl. Proof: Define A'(t)" V --, V ' b y J~t(u, v) = (A' (t)u, v) for u, v E V. It holds that

0 (zI - A(t)) -1 = - ( z I - A(t)) 0-7

-1 d'

(t) (zI - A(t)) - 1

We have

0 =

(zI(zI q-(zI

A(t)) -1

0

A(s))-I 1

(zI-

A(~)) -1A'(t)(zI -

+ (zI - A(t)) -1

A(t)) -1 [A(/~) - A(s)]

-

d(f)) -1 [m'(t)

-

(zI

[A(t) - A(s)]

d9' ( s )

The equality

m'(s)]

-

(zI -

A(s)) -1

d(s)) -1

(zI -

A(s)) -1

(zI - A(s)) -1 d'(s)(zI

-

d(s)) -1

114

3. Evolution Equations and F E M

arises with the following terms" Bl(t, 8)

=

Ao(t) -1/2 ( z I - Ao(t)) ( z I - A(t)) -1Ao(t) 1/2 A9o ( t ) - l / 2 A ' ( t ) d o ( t ) -1/2. do(t) 1/2 ( z I - A ( t ) ) -~ do(t) 1/2 Ao(t) 9 -1/2 [A(t) - A(s)] Ao(t) -~/9

Ao(8) 9 1/2 ( z I - A(8)) -1 ( z I - A o ( s ) ) d o ( s ) -~/2, B2(t, s)

=

mo(t) -1/2 ( z I - Ao(t)) ( z I - A ( t ) ) -1 do(t) ~/2 Ao($) 9 -1/2 [A'(t) - A'(s)] do(s) -'/2

A(8)) -1 ( z I -

Ao(s) 9 ~/2 ( z I Ba(t, s)

=

A o ( s ) ) A o ( s ) -1/2,

Ao(t) -1/2 ( z I - Ao(t)) ( z I - A(t)) -1Ao(t) 1/2 Ao(t) 9 -1/2 [A(t) - A(s)] Ao(s) -1/2. Ao(s) 1/2 ( z I - A(s)) -1Ao(s) 1/2 d9o ( s ) - ~ / e A ' ( s ) d o ( s ) - l / 2 ,

m o ( s ) ( z I - A ( s ) ) -~ ( z I - do(s))Ao(s) -Vg

Thus the lemma has been reduced to the following inequalities:

IlAo(t)-'/~ (zI

- Ao(t)) ( z I - A ( t ) ) -~ Ao(t)l/2ll <_ C,

IlAo(t)-l/2A'(t)Ao(t)-l/21l

<_ C,

< C It - ~1,

IlAo(t) -1/~ [n(e) - A(~)] no(~)-'/=ll

IIAo(t)-'/2 (zI - n ( t ) ) - ' (~I - Ao(e)) no(e)-'/=ll < C,

IlAo(t)

A'(~)] nol/=ll _< C l t - ~ l

~

They are direct consequences of (3.23), (3.25), and the following inequalities:

IA(t)l v,v, ~ C,

C -1 ~

C -1 ~ IIA0(t)Iv, v, < C,

I A'(t)llv, v, <_ C, A(t) - A(s)llv, v, < C I t - sl,

IIA'(t) - A'(.s)l v,v, < C I t - ,s ~. Details are left to tile reader. Now we show the following inequalities as consequences of Lemmas 3.3 and 3.1, where

9 ~ (0,1/2). A ( t ) -O~ A ( t ) -~ (9

(zI - A(t))

-1

0

- ~

(3.60)

<_ C o ,

( z I - A(s)) -1

A ( t ) o f f - - T A ( t ) - ' - A ( s ) o "ff-~A(s)- ~

< C It - s] ~ , < CI~ It -,sl "

(3.61) (3.62)

3.2. A Priori Estimates Proof of (3.60):

115

Inequality (3.26) (with a = 1) gives

l]A(t) ~ [A(t + e) -~ - A(t) -z]

tl -<

and hence (3.60) follows because

ate3A(t)_ ~ ---- 27rzl/r z-~-~c3 (zI -

A(t)) -1

dz

exists. P~oof of (J.61): If H is a positive definite self-adjoint operator in X, its spectral decomposition assures

IIH1/~ (zI - H)-~I _< c for z E E01. Inequality (3.61)follows from (3.59).

Proof of (3.62):

Noting

cOA(t)Z . A(t)_Z ' A(t)Z at~---A(t)-~ = cot we have

A(t)~ ff---iA(t)-~- A(s)~ ff---~A(s)-~ = - ata a(t)~. A(t)_ ~ + ff__~A(s)~.A(s)_ ~

[-~A(t)Z- osOA(s)Z] A(t)-z + CgA(s)3" A(s)-Z [IInequality (3.62) is reduced to

[ff---~A(t)~- ~sA(S)~J A(s)-~[[ < C~ t-s[ ~.

(3.63)

We have assumed the boundedness of A(t) so that (2.18) holds for any v E X. It follows that

8 A(t)Zv =

sin 7r/3 ]i~176Zc3

7r

1

tt -~ (#I + A(t))- v dl~.

If we apply Lemma 3.3, we get

sin7cflTr/o~176

(B-u(t's)A~

(pI + Ao(s))-lv, Ao(t) 1/2 (pI + Ao(t))-lx).

3. Evolution Equations and FEM

116 Inequality (3.63) follows from

( OA(t)~-~

--~sOA(s)~l C

(fOOO#Z3 [Ao(8)1/2

( # I --t-

A0(8))-1 '/)112)1/2

IlAo(t) ~/~ ( . I + Ao(t)) -~ ~[l~ d.

:

It -

sl ~

c~ IIA0(~)~llx. I~llx" t - ~1~

and (3.25). We are able to complete the proof of Lemma 3.2.

Proof of (3.58): We have Oq e_(t_s)A(t) A(t)-ZR~ (t, s) = - A ( t ) - z - ~0qe -(t-~)A(t) - A(t)-Z-~s

=

O -~

_

(A(t)_ze_(t_s)a(t) ) + A ( t ) l - . e_(t_~)a( t)

0__ A + ot ( t ) - "

=

e

-(t-s)A(t)

1 Ot 0 f_ 2m r z-Ze1

27r~

(t-s)z

(zI - A(t)) -1 dz

Jfr Z 1-~e-(t-~)z (zI

--

A(t)) -1 dz + -~ 0 A(t)_ ~ e_(t_s)A(t) 9

1 f~ z-Ze -(t-~)zO (zI - A(t)) - l dz + o__ Ot A (t)-z . e-(t-~)A(t) 2m Ot Therefore, the equality =

16

A(t)-~Rl(t's) - A(r)-~Rl(r's) = E Ze(t'r's)

g=12

holds with

Z12(t, r, 8) =

1 ofr z-~e -(~-~)z [e -
zla(t, r, z) -

1 Jfr 27rz

~e-(~-~)z [0-o-i (z: - A(t)) -1 - ~0 (ZI-- A(t))-lJ d~,

Z14(t,r,z)

=

Z15(t, r, Z)

:

Z'6(t, r, s)

=

0-t0A(t)_ ~ " A(t);~ -~r0 A(r)_ ~ . A(r)f3] A(t)_~e_(t_s)A(t) ~O

A(r)_ ~ A(r);' [A(t)-Ze -(t-s)A(t)- A(t)-Zc-("-s)A(t)],

O A ( r ) - " . A(r) ~ [A(t)-/~e -(r-s)A(t) - A(r)-~ -(r-s)A(r)] Or

3.2. A Priori Estimates

117

Here, we have

,•'0 ~

<

C

# - z e -(r-s)'c~176 (9t - r)# d# #

--

C/3(t-

1")(/--

<

C

p - Z e -(r-s)ac~176 . (t - r ) a d #

8)/5-1

and

IJZ'a(t,

r, s)

:

j~0~176 c~(t- ,-)~(r- s)~-'

Taking the adjoint operators in (3.62), we have

cot (t)-Z " A ( t ) z

A ( s ) - Z " A(s)Z

< Cz

It

- sl

This implies

]lz'4(t, ~, s)JI ~

A ( t ) - ~ . A ( t ) ~ - -~rA(r) -~

IIA(t)-~ll. 9 II~-('-~)A(~)ll s) ~.

_< G(tSimilarly, we have

}I~ at

A9( r ) ~

(t)-~ " A ( t ) z

-
and hence

9

[[A(t)-l~e-(t-s)A< t) _ A(t)-Ze-(r-s)Att)[[

C/~(t- r)(? ~ -- 8) ~-1 hdds by (2.19). Finally, inequality (3.56) is valid for/3 > - 1 and therefore

[IA(t)-Ze-(r-s)A(t) _ A(r)-~e-(~-sla(r)ll <

c~(t-

~)(,~-

~)~-'

follows. The proof is complete. Modifying the proof of Lemma 3.2 slightly, we have (3.54) with/3 = 0 and Zo(t, r, s) satisfying IIZ0(t, ~,

s)ll

<_

c ~ ( t - r)~(r - s) -~

for 0 < ~ < a. Combining this with (2.20), we get IIA(t)U(t, s) - A ( r ) U ( r , s)ll < G ( t

- r)~(r - s) - ~ - 1 + C~(t - r)~(r - s) -~

(3.64)

for 0 _< s < r < t _< T, where/3 E [0, 1] and ~c E [0, a). The proof is simpler and left to the reader.

3. Evolution Equations and F E M

118 3.3

Semi-discretization

As is described in w parabolic equation (3.1) with (3.3) (or (3.4)) and (3.2) is reduced to evolution equation (3.5) with (3.6). In the same way as in the preceding chapter, this equation is discretized with respect to the space variables x = (xl,x2). We triangulate f~ into small elements with the size parameter h > 0 and denote by Vh C V the space of piecewise linear trial functions. As before, Xh denotes Vh equipped with the L 2 topology. The operator norm II " Ilxh,x, is written as I1" II for simplicity. The m-sectorial operator in Xh associated with r h is denoted by Ah(t). Finally, Ph : X ~ Xh is the orthogonal projection. Then, the semidiscrete finite element approximation of (3.5) with (3.6) is given by duh --+ dt

Ah(t)Uh = 0

(0 < t < T)

(3.65)

with Uh(0) =

Phu0

in Xh. Because dim Xh < +oo and Ah(t)'s are smooth in t, they generate a family of evolution operators {Uh(t,S)}o
] Ah(t)Uh(t, S)ll + IlUh(t, S)Ah(S)ll <_ C ( t - s) -1 and

(3.66)

IIuh(t,~) l < c

hold uniformly in h for 0 < s < t < T. Here, the theories Fujie-Tanabe and KatoSobolevskii are applicable. The theory of Kato-Tanabe works also from the argument in w We turn to the error estimate for scheme (3.65). Employing the method of Helfrich, we show

Ileh(t)l x < Ch'2t-' '~011x

(0 < t < T)

for eh(t) = u ( t ) - uh(t), and extend the similar result of w introduced the error operator Eh = Eh(t, s) by

(3.67) For this purpose we

G ( t , ~) = u(t, s) - Uh(t, s)P,~. Obviously,

~h(t) = E,.(t. 0)..0 holds and inequality (3.67) is reduced to

IIG(t, s)ll < c h 2 ( t - s)-'

(o < s < t <_ T ) .

(3.68)

3.3. Semi-discretization Calculations of w equality

119 are valid even for temporally inhomogeneous generators. From the

O [Uh(t, r)PhEh(r, s)] = Uh(t r)[Ah(r)Ph - PhA(r)] U(r, s) Or follows

PhEh(t, s) =

~st Uh(t, r)[Ah(r)Ph

- PhA(r)] U(r, s) dr.

Introduce the Ritz operator Rh(t)" V ~ Vh through the relation (~r

x;, ~ v.).

(3.69)

The equality

Ah(t)Rh(t)v = PhA(t)v

(v E D (A(t)) C Vh)

holds similarly to (1.24). We have

Eh(t, s)

= =

[I - Ph] Eh(t, s) + PhEh(t, s) E~(t, ~) + EX(t, ~) + e~(t, ~)

with

E~(t, s)

=

E~(t, s)

=

E3(t, s)

=

/t

[I - Uh(t, S)Ph] [I -- Rh(t)] U(t, s), Uh(t, r)A~(r) [Rh(t) -- Rh(r)] U(t, s) dr, Uh(t, r)Ah(r)Ph [I - Rh(r)] [U(r, s) - U(t, s)] dr.

It sumces to show that

IIE~( t, ~)ll -< c h ~ ( E s t i m a t e of

t -

8)-1

( ( = 1,2,3).

E~(t, s)

We have shown inequalities (1.38) and (1.42) for v E V N H2(~):

II[Rh(t) - I] ~llv

<

II[Rh(t) - I] ~llx

<

Ch Ilvl ~.(~),

Ch~ll'~ll_,-,~(~).

Therefore, from the elliptic estimate follows that

[IE~(t,s)ll

~_ (I + l Uh(t,s)l I . IIPhll) . l[[I- Rh(t)]A(t)-lll . llA(t)U(t,s)l I ~_ C h 2 ( t - s) -1

(3.70) (3.71)

3. Evolution Equations and FEM

120

Estimate of E~(t, s) We shall show the inequality

ll[Rh(t)-/~h(S)]Vllx <--Ch 2 It- sl ~ IvllH~(a)

(3.72)

for v E V N H2(f~). Then it follows that IIE~(t,s)ll

_

IIUh(t,r)Ah(r)ll

tl[Rh(t) - R~(~)] A(t)-' l[" IIA(t)U(t, s)ll d~ <

c

( t - s)-l+~h~d~. ( t -

= Ch2(t-

~)-~

Ch2(t- s) -1. To show (3.72), we recall the adjoint form .A~('u, v) = .At(v, u), and denote by/~h(t) the 8)-1+ c~ ~

Ritz operator associated with it:

.,4; (,Oh(t)~, ~:h) : A;(v, ~:h)

(,, ~ v, ~,, c v,,).

The inequalities

ll[/~h(t)--/]vlIv~ ChlvllH~(~) and

I][~h(t) -/]vllx ~ Ch21v IH~(nl hold similarly to (3.70) and (3.71), respectively, where v C V N H2(~2). Setting z = [Rh(t)- Rh(s)] v E Vh, we have

llzll

= A, (z,

A, (z,

=

(At - As) ([1 - Rh(s)] v, [~h(t)A(t)*-lz)

=

( A t - Ms)([1-- Rh(S)] V, [/~h(t)- I] A(t)*-lz) + ( A ~ - A s ) ( [ I - R/~(s)] v, A(t)*-lz)

= ( A t - A s ) ( [ I - Rh(s)]v, [/~h(s)- I] A(t)*-lz) + As ( [ I - Rh(s)] v, [A(s) * - 1 - A(t) *-1] z)

:

(,At- ,As)([I- Rh(8)] "u, [il~h(8) - I]

A(t)'-lz)

+ .As ([I - Rh(s)] v, [I - Rh(s)] [A(8) *-1 - A(t) *-1] z)

_<

+ c I I [ / - Rh(s)] vllv I[[1 - ~h(S)] [A(s) * - ] - A(t) *-]] 6' It - sl ~ h 2 Ilvl g=(~) IA(t) *-'zl ,,=(.) + Ch 2 Ilvlln=(~) l] [A(S)*-I -- A(t)*-l] ~11.~(~)-

zll~

3.3. Semi-discretization

121

The elliptic estimate implies

IIA(t)*-x~llH~(~) ~ c Ilzll~ and

II [A(s) *-1 -

A(t) *-1] ZlIH2(a) -< C It - sl ~ Ilzllx.

Inequality (3.72) has been proven.

E s t i m a t e of E3(t, s)

First Case

If V = H~(f~), the duality argument of Helfrich is applicable and inequality (3.22) follows. For Uo E D ( A ( s ) ) =_ D we have

jfst IIUh(t, r)Ah(r)ll.

SO ollx

I11I -/~h(r)] A(t)-l[[

IlA(t)[u(t, 9 s) - U(r, s)] A(s)-lll <_ Co

=

(t - s)-lhg(t - r)~

- s)-~

IlA(s)uollx 9 ds

IIA(s)u01 x

Ch 2 IIA(s)uollx

with 0 E (0, c~). We can show

for g = 1, 2 similarly, from the second estimate of (3.21). (Details are left to the reader.) We have

IIEh(t, s)A(s)-~ll < Ch 2.

(3.73)

Now, the semigroup property of evolution operators implies

Uh(t, r)Uh(r, s) = Uh(t, s)

and

U(t, r)U(r, s) = U(t, s)

for 0 _< s _< r < t _< T. The identity

Eh(t, s) = Uh(t, so)PhEh(So, s) + Eh(t, so)U(so, s)

(3.74)

follows with so = (t + s)/2. The second term of the right-hand side of (3.74) is estimated as

IIE~(t,~o)U(so, s)ll

<_ IJE~(t, so)A(~o)-~ll'llA(so)U(so, <

C h 2 ( t - s) -1

s)ll

3. Evolution Equations and F E M

122 by (3.73). On the other hand, (3.69) gives [[Uh(t,S)PhEh(So, S)II =

[IU~(t, so)Ah(So)Rh(So)A(so)-XEh(So, s)ll

< [Iuh(t, so)Ah(So)Ph [Rh(S0) - I]A(so)-lEh(So, S)] + ]lUh(t, so)Ah(so)PhA(So)-IEh(So, s)ll C(t- s) -1 {] [/~h(SO)- I] A(8o)-1 I 9 IIE,~(~o,s)ll +llA(so)-lEh(so, s)[I}. Because IIEh(So, s)ll ~ c follows from (3.66), inequality (3.68) is a consequence of

]A(t)-lEh(t,s)l

<_ Ch 2.

(3.75)

To prove this inequality, let

U(t, s) = U ( T -

s, T -

t)*

(Ih(t, s) = Uh(T - s, T - t)*.

and

Then,

{U(t,S)}o
{U~(t,S)}o
and

are nothing but the families of evolution operators generated by A(t) - A ( T t)* in X and Xh respectively. The relation

Ah(t) = A h ( T -

l~h(t, s) = l~l(t, s) - [Ih(t, S)Ph = E h ( T - s, T - t)* follows. Because

I1E,,/,sIAl,/ 11 is proven similarly to (3.73), we get (3.75) as

I[A(t)-lEh(t,s)[I E s t i m a t e of

Eta(t, s)

= ]lEh(t,s)*A(s)*-l][

<_ Ch 2.

Second Case

If V = H I ( ~ ) , D ( A ( t ) ) varies as t changes. We cannot expect the estimate

IIA(t)U( t, s)A(s) -1]1 <- C or IIA(t)[U(t, s) - U(r, s)]A(s) -1 II -< Co(t - r)~ In this case, we apply the method of telescoping as

5

(t- ~)E,~(t, ~) : ~ F~(t, s) g=l

- s) -~

t)* and

123

3.3. Semi-discretization

with the following terms:

fs t (r - s)Uh(t, r ) A h ( r ) . Ph [I - Rh(r)] [U(r, s) - U(t, s)] dr,

F~(t, s)

=

F~(t, s)

=

(t - r)Uh(t, r ) A h ( r ) " [Rh(s) -- R~(r)] [U(r, s) - U(t, s)] dr,

F~(t, s)

=

(t - r)[Uh(t, s ) A h ( s ) -- Uh(t, r)Ah(r)]

f'

Ph 9 [I - Rh(s)] [U(r, s) - U(t, s)] dr, (r -- s)[U(r, s) - U(t, s)] dr,

F2(t, s)

=

--Uh(t, s ) A h ( S ) " Ph [I -- Rh(S)]

F2(t, s)

=

Uh(t, s ) A h ( s ) . Ph [I -- Rh(S)] (t -- s)

[U(r, s) - U(t, s)] dr.

We have to show that [IFeh(t, s)l I _< C h 2

(g = 1, 2 , . . . , 5).

(3.76)

Recall that Kato-Tanabe's generation theory is applicable to (3.5). Combined this with the elliptic estimate we can show the following for 0 _< s < r < t _< T, where 0 _
IlU(t, 8) - U(r, 8)IlL2(FI),H2(~2) ~_ C k { ( t - T)~(T -- 8) -~-1 -1- ( t - T)~(T -- 8)-1} , IlUh(t,r)Ah(r) -- U , ( t , s ) A ~ ( s ) l l

<_ Ck { ( t - r ) - ~ - Z ( r - s) ~ + ( t -

/s'

[U(t, s) - U(r, s)] dr

I[

L2(f~),H2(f~)

(3.77)

r ) - l ( r - 8)a},

(3.78) (3.79)

< C.

Those inequalities are proven in the following way. Proof of (3. 77):

Take u0 E L2(f~) and set u(-, t) = U(t, S)Uo. We have B(x,t,D)u(x,t)

=0

on 0f~ where 13 = O/Ovc + ~. This implies B ( x , t, D) (u(x, t) - u(x, r)) : [B(x, t, D) - B ( x , r, D)] u(x, r)

there. Similarly, we have

[A(t)U(t, s) - A(r)U(r,

s)] uo(x)

+ [s

r, D) - s

t,

D)] u(x,

r)

3. Evolution Equations and F E M

124

in f~. The elliptic estimate gives

II[U(t, ~) - u(~, ~)1 ~olIH~(~) -- I1~(', t) -- ~(., ~)11,~(~) <_ C I~(', t, D) (u(., t) - u(-, ~))11~(~) +C ILL(', t, D) (u(., t) - u(., ~))11,,,~(o~) <_ C I I [ A ( t ) U ( t , ~) - A(,~)U(r, ~)] ~o11~(~) +C I [L(', ~, D) - s t, D)] u(., ~)11~(~) +C lilt3(', ~, D) - B(-, t, D)] u(., ~) I1,,,,~(o~) <_ C I I A ( t ) U ( t , ~) - A ( r ) U ( , ' , ~)11" luoll~(~) + C ( t - ,~) I1~(', ~)11~(~) <_ C IIA(t)U(t, ~) - A(,~)U(, , ~)11 I1'~o11~(~) + C ( t - ,~)(~- ~)-' I1~o11~(~) Inequality (3.64)implies (3.77).

[]

Proof of (3. 78)" This is nothing but the adjoint form of (3.64) applied to A(t) = Ah(t) and U(t, s) = Uh(t, s). [] Proof of (3. 79): Letting u(., t) = U(t, S)Uo, we have s

s, D) (u(x, t) - u(r, x)) = [s

+ [s

x, D) - E(s, x,

s, D) - s

t, D)] u(z, t)

D)] u(z, r) + [A(t)U(t, s) - A(r)U(r, s)] to(Z)

in t2 and B(x, s, D) (u(x, t) - u(x, r)) : [B(x, s, D) - B(x, t, D)] u(x, t) + [B(r r, D) - B(x, s, D)] u(r r)

on cgf~. The elliptic estimate now gives [U(t, s) - U(r, s)] no dr

f'

H2(U)

<_ C(t - s)II[E(., s, D) - s

+C

I[s

r, D) - s

=

('u(-, t) - u(., r ) ) d r

t, D)] u(.,

t)JlL~(~)

H2(~)

s, D)] u(-, r)llL2(U ) dr

+C(t - s)I [~(., s, D) - B(., t, D)] u(-, t)ll,,/~(on) +C <

C ( t - s) 2

[A(t)U(t, s) - A(r)U(r, s)] uo dr

Ji'u(-, t)llu~(n) + C

(r -s)tl'u(,

II

r)liu~(~)

L'~(n) dr

~)11 ~o11~(~) + C IIU(t, ,~) - I I. I1~,o11~(~) <_ C ( t - ~) I uoll~,(~) + C I uoll~(~) ___C I1,,,o I~(,) 9 +c(t

- ~)IIA(t)U(t,

3.3. Semi-discretization

125

The proof is complete. Now the proof of (3.76) is given as follows. First, with/3 E (0, 1) we have

IIF ( t, )ll -< <

/s

- s)Ilgh(t,

IIU(~. ~) - u ( t . ~)l ~ ( ~ ) , ~ ( ~ ) dr

I9I I - R~(~)ll.~(~).~(~)"

C~h 2

8 ) ( t - T) -1

(r9{( t -

<

r)&(r)l

T)~(T -- 8) -~-1 + ( t -

T)~(r -- s) -1 } dr

Ch 2

and

jfs t (t - r)IIUh(t, r)Ah(r)ll

~

IIFZ(t, s)ll

U(t, ~)llL=(~),H~(~)dr

IIRh(~) 9 - Rh(~)l H=(~),L=(~)" IIU0", ~0 -

_< c ~ h ~ fS t (~ - ~) {(t - ~)~(~ - ~)-~-~ + (t - ~)~(~ - ~)-1} a~ <

C h 2.

Next, with 0 < "i' < ~ _
_<

r) l U h ( t , r ) A h ( r ) -- Uh(t,s)Ah(s)l]

(t-

III 9 -

R"(S) IIH=(~),L=(~)"IIU(t,

~) - U(~', ~) IIL=(~),H=(~) d,"

( t - T) { ( t - r ) - l - ~ ( r - 8) ~ Jr- ( t - r ) - l ( r -

C~h 2

8) ~}

{ (t - T)7(r - 8) -1-3' -Jr (t - T)m(r -- S) -1 } dr <

C h 2.

W i t h / 3 c (0, 1), we have

I[F2(t, )ll

s)&(s)ll.

Ilun(t,

III-

Rh(S)IIH=(~),L~(~)

9 IIg(t, s) - U(r, S)[IL2(a),H:(a)" (r -- s) dr

<_

C~(t- s)-~h 2 9

<

{( t -

~)~(~ - ~ ) - ~ - 1 + ( t -

~)~(~-

~ ) - 1 } (~ _ ~) d~

C h 2.

Finally, we have [IF2( t, s)ll

_<

IUh(t,s)&(s)

9 <

C h 2.

liRa(s) 9 - IllH~(~),L~(~)

[u(~, ~) -

u(t,

~)] d~

L2(~"/),H2(n)

3. Evolution Equations and F E M

126 Summing up those estimates, we obtain

IlE2(t,~)ll <_ ch~(t_ ~)-, All the proof for inequality (3.68) is complete.

3.4

Full-discretization

Fully discrete approximations are obtained by discretizing the semidiscrete equation

duh dt

+ Ah(t)Uh = 0

(0 < t <_ T)

with

Uh(O) = Phuo

(3.80)

in the time variable t. In the present section, we adopt the backward difference method with the mesh length r > 0 satisfying T = N r . T h a t is,

U~h(t + r) -- U~h(t)

+ Ah(t + r)u,~(t + r) = 0

(t = n r )

(3.81)

in Xh with

~,~(o) = P~,o, where n = 0, 1,-.- , N. Thanks to (3.14), tile scheme is uniquely solvable and

e~h(t) = u h ( t ) - U,;(t) denotes the error, where t = nr. we shall derive

Id,(t)llx _< c~-t-' luol x and extend a result in w

(3.82)

Combining (3.82) with (3.67), we obtain

u(t) - u;(t)l x < C (h? + r) t-~ I 'uo Ix

(t = n r ) .

(3.83)

Let t,~ = nT and set

U[~ (tn, tj) =

{ ( / h + TAh(tn)) -1 (Ih + TAh(tn-1)) -1"'" (Ih + TAh(tj+l)) -1 Ih.

We have ~,;(t) = ui:(t, o)&~o

(t = t,,)

and

,,,,, ( t ) = u,, ( t, 0)P,,,~,o.

(~ > j) ( ~ = j).

3.4. Full-discretization

127

Drop the suffix h for simplicity of writing. Because of (3.80) and (3.81), we have for t = tn that

e ~(t + r) - e r(t)

tnt-r

f f

=

[A(t + r)u r(t + r) - A(r)u(r)] dr

at

=

t+r

[A(t + r)u(t + r ) -

A(r)u(r)] d r - r A ( t + r)e" (t + r).

,It

Hence

er(t -+- 7) = (1 + r A ( t + "r) )-l er (t) + (1 + r A ( t

+

T)) -1

f

t+r

[A(t + r)u(t + r ) -

A(r)u(r)] dr.

Jt

Because of e ~ (0) = 0 we obtain

e ~(tn)

-- E ~(tn)Puo =

(I q- 7A(tn)) -1 (I + "rA(tn_l)) - 1 . . . (I + TA(tk)) -1 k--1

-1

[A(tk)U(tk, 9 O) - A(r)U(r, 0)] drPuo,

(a.s4)

Er(t) being the error operator: E ,(t) = u(t, o)P - u ~(t, o ) Inequality (3.82) is reduced to those on the stability of the approximate operator U r (tn, tj+l) and the smoothness of the original one A(t)U(t, s) - A(r)U(r, s). We have proven Lemma 3.2 for the latter. As for the former, we have the following lemma, proven similarly to the continuous case: L e m m a 3.4. For each/3 E [0, 4/3), the inequality

II

tj)A(tj+l)Zll <__Cz(tn - tj) -~

(a.85)

holds true. Proof." Taking adjoint form reduces inequality (3.85) to I A(tn)ZU~(tn, tj)l] <__Cz(tn

-

tj) -~,

of which continuous version is proven in w

IIA(t)ZU(t, s)ll<_ Cz(t- s) -z All we have to do is to trace computations

in the context of discreteness.

(3.86)

3. Evolution Equations and F E M

128

In fact, inequality (3.26) holds for A(t) = Ah(t) uniformly in h as is described there. Identity (3.30) holds for A(t) = Ah(t) so that we obtain

U'(tn, tj) - (1 + rA(tj)) -(n-j) =

~

[(1 + ~-A(t~))-(~-k)U'(tk, t j ) - (1 + 7A(t~))-(n-k+l)U'(tk_x,tj)]

k=j+l

=

~

(1 +7-A(tn))-(~-k+l) [(1 +~-A(tn))- (1 +~-A(tk))]U'(tk, ty)

k:j+l

= T ~ (1 + rA(tn)) -('~-k+~) [A(tn) - A(tk)] U~(tk,tj) k=j+l = ~p=l

(1 + 7A(tn))-(n-k+UA(tn)l-PPD(tn, tk)A(tk)PPVr(tk,tj).

~

(3.87)

k=j+l

Given operator-valued functions fit = Ke(tn, tj) with t~ = 1, 2 on

D r={(t~,tj) I N>_n>_j>_O}, we define K = K1 .r /42 by n--1

(K1 *~ K2) (tn, tj) = ~- Z

Kl(tn, tk)K2(tk, tj).

k=j+l

Furthermore, we set

WT(t~,tj) Y~(rn, tj)

= Ur - (1 + TA(t~)) -(~-j) = A(t~)qPWT(t~,tj).

(3.88) (3.89)

Then, equality (3.87) reads: m

Yq = E p=l

H q~p , ~- U + Yq,~O,

(3.90)

where

Hq~p(t~, tj)

= A(tn)I-pP+qP(1 + ~-A(t~))-(~-J+l)D(t~, tj),

(3.91)

m ,0

=

,p

,- 1

(3.92)

p=l

with

Yp~_l(t,,,tj) = A(tn)PP(1 + TA(tn)) -(''-j)

(3.93)

Note the elementary inequality (3.94). The desired inequality (3.86) can be derived in a similar way to that in w

3.4. Full-discretization

129

Details are left to the reader, but the following inequality is worth mentioning, where

O < a
B N(a, b) - -~1 k=l

1

- -~

< S(a, b) =-

/o (1

-

-

x)a-lxb-ldx.

(3.94)

In fact, f(x) = (1 - x)a-lx b-1 is monotonically increasing in [0, 1) when b _> 1 _> a, while f(x) is convex in (0, 1) when a, b _< 1. Those facts imply (3.94). [] We are able to give the following.

Proof of (3.82):

E~(tn) =

The operator E ~ in the right-hand side of (3.84) splits as

n~k

(1 + ~-A(tn)) -1... (1 + TA(tk))-lA(tk)[e -tkA(tk)- e -rA(tk)] dr

E k=l

-1

(1 + rA(tn)) -1... (1 + rA(tk))-lA(tk)~Z~(tk, r, O) dr

+ k=l

=

-1

(1 + TA(tn))-lA(tn)

[e-tnA(t~) - e -rA(t~)] dr --1

+

Ur(tn, tk_l)A(tk)[e -tkA(tk)- e -ra(tk)] dr k=l

-1

~

+~

k=l

U'(tn, tk_l)A(tk)ZZz(tk, r, O) dr.

(3.95)

-1

Inequality (2.19) is applied to A = Ah(r) uniformly in h. We have for n > - 1 that

[IA(r)~[e -tA(~) -~-~A<~>]II C (t- s)s -n-l,

(3.96)

where 0 < s _< t < oc. Supposing n > 2, we can estimate the first term of the right-hand side of (3.95) as

[1(1 + TA(t~))-lA(tn)ll

~

n

lie -t~A(t~) - e-~A(t~)ll dr

--1

CT

( t n - r)r-~dr <_ CT-1T2(tn-1

-1 --1

~ Crt-

3. Evolution Equations and F E M

130

Next, by Lemma 3.2, the third term of the right-hand side of (3.95) is estimated as

k~l

Ilg~(t~,t~-l)A(t~)z[I

--1

< C a k1= ~

IIZz(tk, 9 r,O)ll dr

-, (tn - tk-l)-Z(tk -- r ) r - l + Z d r

n

___ c~ ~ ( ~ -

k + 1 ) - ~ - ~ ( k ~ ) ~-'

k=l n = c~

Z(~

- k + 1)-~k ~-1

k=l Inequality (3.94) now gives that

~~ IIg'(t~,

tk_l)A(tk)Zll ]lZz(tk, 9 r, O)l I dr <_ CT. k=l The desired inequality (3.82) has been reduced to

fi'

,11

To see this, take/3 C (0, 1/3). We get

n En-1 o~tl k U~(tn, tk_l)A(tk) k=l

[e - t k a ( t k )

- e -~A(tk)] dr

-' n-1( n = E k + 1)UT(t~, tk_l)A(tk) 1+'. ~tlk A ( t k ) - ' [e -tkA(tk) --e -rA(tk)] dr k=l -1 n-1

+ E(k

- 1)UT(tn, tk_l)A(tk) '-~

k=2 9

A(tk) ~ [e -tkA(tk) - e -rA(tk)] dr.

(3.98)

-1

The first term of the right-hand side of (3.98) is estimated by Lemma 3.4 and (3.96) as

n-1 ~-~.(n-k+ 1) llUT(t,~,tk_~)A(t~)'+Zll. ~t.lk IlA(tk) -~ [et~A(t~)--e-~A(t~)]l I dr k=l

-1

_< Cr Z ( r t - k + 1)-'T - 1 - ' ~t.lk (tk -- r)r'-idr k=l -1 n-1

rt-1

< C/3 Z ( ~ 9 , k=l

]~:-J-- I)-~qT -1-/~ T2(~T)/9-1 9

rt

:

Ca E ( n k=l

If + 1 ) - ' k '-1 _< C.

131

3.5. A l t e r n a t i v e A p p r o a c h

Similarly, the second term of the right-hand side of (3.98) is estimated as

n-1 E(kk=2

1)IIU~(t~,tk_l)A(tk)l-~ll. _< C 0

n-1

( k - 1)(tn - t k - ,

k=2 rz-1 _< c , ~ ( k k=2 n-1 < Cr E ( k k=2

/i k -1

)-1+/~

IIA(t,)~

9

[~-~(~)- ~-~(t~)] II d~

(tk - r ) r

-/~-1

dr

-1

- 1 ) ( ~ - k + 1) -1+~ ~-1+~. 9 ~ . ((k - 1)~) -~-~

- 1)-r

k -+- 1) r

< C.

The proof is complete.

3.5

Alternative

Approach

Inequality (3.83) is proven in a different way by the energy method. In this section, we describe the arguments of M. Luskin and R. Rannacher for the semidiscrete approximation and M.Y. Huang and V. Thom~e for the fulldiscrete approximation, respectively. A Priori Estimates

First, we show some estimates concerning the solution of the equation du d--7 + A ( t ) u = f (t)

with

(0 < t < T )

u(0) = ,to

(3.99)

in a Hilbert space X over R. Here, A ( t ) is an m-sectorial operator associated with a bilinear form At( , ) on V x V through a triple of Hilbert spaces V C X C V*. We suppose

I.,4t(,< ~)1

<

C1 Ilu Iv IIv Iv

and

.a~(~, ~) > 5 II~ll~-

(3.100)

for u, v E V, where C 1 > 0 and 5 > 0 are constants. It is also supposed to be smooth so that oq A (u, v)

and

02

exist and satisfy

IAt(,

and

A~(,~, ~)1 ~

c

I1~11~Ilvllv

132

3. Evolution Equations and F E M

for u, v C V. Problem (3.99) admits the weak form (Ut, ?J) + .At(u , v) ~- (f(t), v)

(3.101)

for v C V, where ( , ) denotes tile inner product in X. This implies (Utt , 72) nt- .At(ttt, v) -- - A t ( u ,

v) -nt- (It(t), v).

(3.102)

/0

(3.103)

We have the following. P r o p o s i t i o n 3.5. The inequality

Ilu(t)ll 2x +

/0

Ilu(s)llv~ ds _< C I1~oll ~x + c

IIf(s)ll~.ds

hold, fo~ th~ ,ol~t~o~ ~ = ~(t) of (S.99).

Proof: Putting v = u(t) in (3.101), we h~ve ld

2 dt Ilu(t)ll~ + 5 Ilu(t)ll~ _< Ilf(t)llv. Ilu(t)llv by (3.100). Then, Schwarz's inequality gives

-

and

d dt Ilu(t)ll2x + l u(t)llv2 < C Ilf(t)l 2v. -

_

inequality (3.103) follows.

[]

Letting .A;(u, v) = .At(v, u) and A ~ = (.At + .At)/2, we suppose that

v) + (B(t)u, v)

.At(u, v) = A~

(3.104)

holds with a bounded linear operator B(t) : V ~ X. This assumption is satisfied if .At(, ) is associated with a second-order elliptic operator with real coefficients. We have the following. Proposition

3.6. The inequality

ilu(t)ll 2v +

/0

Ilu~(s)llx2 ds _< C Iluoll 2v + C

/0

f(s)l

x

ds

holds for the solution u = u(t) of (3.99). Proof: Putting v = ut in (3.101), we have

Ilu~ I~ + A~(u, u~)

= (f,

Ut)

< --

1 2 1 ~ II/11X + 2 Ilu~ I~

"

On the other hand, equality (3.104) gives

1d

1

.At(U, Ut) = -~--~.At(U, U) -- = A t ( U , U) nrZ

1.B.t.u,( ( )

Ut)

Q

(3.105)

133

3.5. Alternative Approach

Those relations imply lutll x~ + ~dA , (u, u ) <_ C I/ll x~ + c I1~ Iv. Then, inequality (3.1o5) follows from (3.100) and (3.103). P r o p o s i t i o n 3.7. The inequality Ilu,(t)l x~ +

/o

I~,(s)llv~ ds _< C Ilu~(0)ll x2 + C l'~o I.~

(If(s)I~,. + II/~(~)11~v*) ds

+ C

(3.106)

holds for the solution u = u(t) of (3.99). Proof." Putting v = ut in (3.102), we have

ld 2 dt I1~'11~ + 61~,11~, < -<

c II~llv I1~ Iv + f~llv. I~1 v 1

-~11~ ~v + C' I1~ Iv~ + C IIf~llv* 9 2

This implies

Ilu~(t)ll ~x +

/o

Ilu~(sDllv~ds_< II~,(o)ll~ + c'

/o

II~(~)ll ~vds +

c

/o

II/t(s)ll~.

ds.

Then, inequality (3.106) follows from (3.103). P r o p o s i t i o n 3.8. The inequality

/0

s ~ Ilu,(s)ll~, ds _< C Iluoll.~ + c

ji

(ll/(s)[Iv. + II/~(s)ll 2v.) ds

holds for the solution u = u(t) of (3.99). Proof: Putting v = t2ut in (3.102), we have

ld 2 dt (t2 I]utl]~.) + 5t 2 Ilu, ll ~v < - ? A , ( ~ , u,) + t I1~11x~ + t2(ft, ut) -

-

v + c II~llv + t II~ll~ + ~t ~ II~,ll~, + c II/~ll 2V. 9

This implies d~ (t2 I[ut[ ~) +

I[ut][y _

g + t II~,llx + c IIf~ll~.

(3.107)

3. Evolution Equations and FEM

134 and the inequality

/o

s ~ Ilut(s)ll*v ds _< C

/o t (ll~(s)ll~

+ s lut(s)12x + lift(s)Iv*) ds

(3.108)

follows. Next, putting v = tut in (3.101), we have

t i1~,11~x+_did (tAt(u, for c > 0. The inequality

jot

II~t(~)ll ~ d ~ <_~

u))

--

At(u, u) -t- .At(u, u) -t- At(tut, u) -t- (f, tut)

-<

~t ~ll~t ~v + c~ II~llv + c~ IIf I~.

/~t

~ II~t(~)llv~ d ~ + C ~

jot (11~(~)11'v +

If(~)ll~.)d~

follows. Taking e > 0 small enough and combining this with (3.108), we have

/o s21lut(s)ll~ds c /o { Ilu(s)ll~ + II/(s)I~. + IIf,(s)I~. } ds. _<

Then, inequality (3.107) follows from (3.103). Proposition t llu,(t)ll~ +

3.9. The inequality

i

X

t

II~.(s)ll ~ ds <

c ilu,(o)ll x~ + c II~ollx~ + c

/o'

(llf~(~)llx~ + IIf(~)llv.) d~ (3.109)

holds for the solution u = u(t) of (3.99). Proof." Putting v

~-- Utt

in (3.102), we have

IlUttll 2X -Jr-,At (ut, utt) : -At (u, utt) + (ft, tttt).

(3.110)

Here, note that

12ddt.At(ut, ut) = < - At (.tz, tttt)

.At (ut, uu) + ~1 At (ut, ut) + 1 (But,

r

?ztt) + C I1..,11~

1

Utt)

2

d 9 ~-- - d--iAt (11., ?zt) -~- At (.~z,"~l.t) + At (ut, ut) <

-

d dt

A,(,~, .,~,) + c I1,,, } + c

ul ~v ,

135

3.5. Alternative Approach and 1 We obtain

d d &(~, ~,,) -< C I~,1 ~ + CII~I ~ + CIIf, ll~ u.ll~: + aTA,(~,, u,)+ ~7 Multiplying t now gives that

'(

)

t lluttll2x + -~t A~(u~, u~)+ At(u, ut) -< c(t +

1)(llut]l

2v + I~1 ~ +

Ift I~)-

This implies

tA~(~,~) +

/o

~ll~,(411~ <_ c

/o

(ll~,(~)llt + 11,,(~)115 + IIf~(~)ll~) d~.

Inequality (3.109)follows from (3.103)and (3.106). Semidiscrete Approximation Let fZ C R 2 be a convex polygon, and A t ( , ) a bilinear form on V x V with V = H~(f~) (for simplicity) induced from a second order elliptic differential operator. The associated m-sectorial operator in X -- L2(f2) is denoted by A(t). We take the evolution equation

du + A ( t ) u = O (O < t < T) dt

with

u(0)=u0

with

uh(0) = PhUo,

and its semidiscrete finite element approximation

duh d--T- + Ah(t)Uh = 0

(0 < t < T)

described in w and w respectively. Now we develop the error analysis by the energy method. Recall t h a t Rh = Rh(t)" V ---+Vh denotes the Ritz operator associated with At( , ):

At(Rh(t)v, Xh) = At(v, Xh)

(v e V, Xh r Vh)

(3.111)

We have the following. L e m m a 3.10. Any v = v(t) c C 1 ([0, T] --+ V) satisfies the inequality

Ila~(v- nhV)llH,(~) <_ Ch ~-j (llv g~(~)+ IIV~IIH~-,(~>) fork

1,2 and j = 0 , 1 .

(3.112)

3. Evolution Equations and F E M

136

Proof." Equality (3.111) implies

.A~(a,(v - R,,v), x~) = -A,(,, - R,,~. x~)

(3.113)

for Xh E Vh, and hence

a l a~(v - R~v)ll~

~

.At (oqt(y - R h V ) , Ot(v -- f~h'U))

--

.At (Ot(V -- R h V ) , Otv -- Xh) -- A t ( v -- R h V , Xh)

___ c (lla,(v follows. T h e results in w

Rhv)liv Ila~v

-

- xhll,,,. + IIv - Rhvllv

Ilxhllv)

are summarized as

II(Rh(t)-

1)v g,(n) <--

Chk-Jll~llH~(n)

(3.114)

for k = 1,2 and j = 0, 1. Taking Xh = Rh(t)Otv(t), we get (3.112) for j = 1 and k = 1,2. The remaining case j = 0 is obtained by the duality a r g u m e n t similarly, in use of (3.113) and the elliptic estimate of A(t). It is left to the reader. [] Suppose u0 E V, and take the solution Uh = gh(t) E Xh of duh

dt

+ Ah(t)gh = O (O <_ t <_ T)

~(0) = R~(0)~0.

with

We have the following. Lemma

3.11.

The inequality

Z

~ II~h(~)ll~-d~ ~

Ch~l ~o11~"

(3.115)

holds for -eh = u -- Uh. Pro@

We have

(9~,,, x,,) + A~(~,,, xh) = 0 for Xh E Vh. Taking Xh = Rh(t)-eh(t), we get ld

2dtll~hll~c+ A,(~h,~h)

=

(0,~h, (1 - Rh)-~h)+ At (eh, (1 -- Rh)~h)

=

(ateh, (1 - Rh)'U) + .At (-eh, (1 -- Rh)'U)

(

<_ c~ h~ I1,~11~(~) + Ila,~,~llx

) +cllehllv

for c > O. This implies

d -dt

1~,,112 x + -~hl ~ _<

Ch ~

( I1~11~,-,~(~)

+ Ila~,,llx

)

137

3.5. A l t e r n a t i v e Approach

by (3.100), and hence

/0

II~h(~)ll ~V dx

<

Ch ~

<_ c h ~

I1~(~) .:(~) + IIO~h(~)ll

ds + II~h(o)llx

II~(s)

+ ch ~ ~ol follows. The elliptic estimate gives

Ilu(s)ll/_z~(r~) < c IIA(~)~(s)Ix : c Ilu,(~)l x, while inequality (3.105) of Proposition 3.6 is valid for u = Uh uniformly in h. The righthand side is further estimated from above by ch ~

(11~ol ~ + IIRh(0)~oll~) <

Ch ~

I1~o

and the proof is complete.

[]

If one assumes the inverse assumption, L 2 orthogonal projection Ph " X ~ X h satisfies

llPhllv_v <_ c

(3.116)

by Proposition 1.5. Then, the inequality

fo ~ II~h(~)tl ~V d~ < is proven in a similar manner, where eh = u assumed in the following lemma.

Ch ~

I1~o11~,

Uh. However, inequality (3.116) is not

L e m m a 3.12. I f uo E V A H k - J ( ~ ) we have

~0 t II~h(~)ll x~ d~ <-

c t l - J h 2k

Iluo

2

,

(3.117)

9V)

(3.118)

where k = l , 2 and j = 0 , 1 . Proof:

Given t > 0, we take the backward problems (v, O~w) - A~(v, w) = (v, e~(s))

(v

with w(t) = 0 in X and

(x~. 0 ~ , )

- As(X,. ~ , ) = (x~. ~,.(~))

(x~ e v~)

with wh(t) = 0 in Xn, respectively. Those problems have unique solutions w = w(t)

9 C O([0, t), D (A(t)*)) A C O([0, t ] , X ) n C ~ ([0, t ) , X )

3. Evolution Equations and F E M

138 and

~

= ~(t)

~ c ' ([o.t]. x h ) .

respectively. In spite of the inhomogeneous terms in the right-hand side, the argument in the proof of Lemma 3.11 is applicable with the time variable t reversed. A quick overview reveals that

/o'

w(~) - w~(~)l ~. d~ _< C h ~

Combining this with inequality (3.105) implies that

L

' (llo~(~)-

<-

(21a, w,,(~) ~x + 2 IIO, w(~)llx~ + h -~ IIw,.(~) - w(~)ll~) d~

< c

--

I~,(~)1 ~ d~.

h-2 II~(~) - ~ , ( ~ ) l l ~ ) ds

o.~,.(~)+

/o' 1'

/o'

Ile,(s)ll ~ ds.

(3.119)

X

Taking v = eh(s) in (3.118), we have

II~,,(s)ll%

=

(e~(s),O~w(~)) - A~(~,,(s),w(s)) {(~,,. o ~ ( ~ - ~ , , ) ) - .4s(~. ~ - w,,)} + ( ~ , , . O s ~ , , ) -

..4~(~,,.w,,).

Here, eh = ( u - RhU)+ (Rhu- uh) and

(~,,. o~(w,,- ~)) - .4~(x,,. ~ , , - ~) = o holds for Xh E Vh. It follows that

I~,(~)11~

=

{(~-

~,,~. o ~ ( ~ - ~ , ) ) - A s ( ~ - R.,,~. ~ d + ~ (~,,. ~ ) - ( o ~ . w~) - .4~(~,,. ~ ) .

~,,)}

Furthermore, the equality

(O~eh, Xh) + A~(Ch, Xh) = 0 holds for Xh E Vh. W e have

I1~(~)11 ~x = ( ~ - R,,,~, o ~ ( w - w , , ) ) - A ~ ( ~ - R ~ , ~ -

d

w,,)+ y~ (~,,, ~,,).

In use of e,,(0) = wh(t) = 0, we h a v e

/0'

II~,,(~)ll ~X ds = _

/o' ]o'

( ( ~ - ~,,~,, a~(w - w , , ) ) -

~

A ~ ( u - R,,~, w - w,,))d~

(llOs~ - a~,,,,,, ~ + t,, -~ w - w , ,

+ c~

/0'

I~)d.~

(I ~ - ~,,,~11~ + h ~ I1,~- ~,,,~11~)d~

(3.120)

139

3.5. Alternative Approach

for c > 0. We obtain

/o

I I ~ ( ~ ) l l ~ d ~ -< C

/o

(ll**-Rh~ll ~~ + h ~ I1~- R ~ I I ~ ) d ~

by (3.119)and (3.120). Inequality (3.117)follows from (3.114). As is described in w

the following theorem implies Ileh(t) x <-- Ch2t-1 Iluollx

by Helfrich's duality method. T h e o r e m 3.13. If Uo E V r~ H 2 (f~), the estimate le.(t)llx < CA ~

luo H~(~)

(3.121)

holds true. Proof." Putting = [(t)= Rh(t)eh(t)= Rh(t)u(t)-

Uh(t) E Vh,

we have ({t, ~) 2t r

~)

--

(Ot(t~hU), ~) -- (OtUh, ~) Jr- r

=

(at(Rhu), ~) - (atu, ~) -(a~(~ - R~), ~).

:

~) -- At(Uh, ~)

Multiplying t implies

1 d (t I1{11~)

2 dt

1 t~t,(~, {) = ~ I1{11~ -

+

t(at(u

-

Rhu)

~)

and hence

t ll{(t)ll~: < c --

J/o'

I1{(~)11~ d ~ + C

f'

~ Ila~(u- R~u)(~)lt ~ d~ X

follows. We obtain

t Ileh(t)ll 2X -< t l l ( u - Rhu)(t)ll~: + C + c

/o

Ile~(s)ll~ d~

I I ( u - R~u)(~)ll~ d~ + C

jo

~ Ila~(u- R ~ ) I I ~ d~.

Here, Lemma 3.10 applies. We get

t lleh(t)ll 2X <- C foot Ileh(s)llx2 ds + Cth 4

{

max [[U(S)IIH2(~) + O
)}

8 [[U(S)IIH2(~) + [[Ut(8)[[ 2u~(n) ds

140

3. Evolution Equations and F E M

For the right-hand side we have

fo' I1~(~)11~ d~ X

< --

C t h 4 I1~o11~(~)

by (3.117), and also

I~(~) IH~(~) _< c II~oll.~(a) and

Jo"(

u(~)ll ~

)

~

d~ < C I ~oll ~

by Propositions 3.7 and 3.9, respectively. Inequality (3.121) now follows. Full-discrete

Approximation

The energy method is also applicable to the error analysis for fully discretized problems. Here we study the backward difference finite element method,

~;(t + r - ~ ( t ) T

+ Aj~(t + T)U "(t h + ~) = 0

with

~;~(o)

=

Phi0,

w h e r e t = t n f o r 0 _ < n < N. We drop the suffix h, and the error is denoted by e(t) = u ' ( t ) - u ( t ) .

The inequality

L e m m a 3.14.

n--1

t 2 Ile(t)ll x2 < - c~ -2 I~011x~ + Cr ~

IIe" (tj)llv.2

(3.122)

j=l

holds for t = tn. Proof:

Given v = v(t), we put

Otv(t) = v(tj) - v ( t j _ l ) T

for t = tj with j _> 1. We have (ut, v) + At(u, v) = 0

and

(O~u~,v) + A~(u ~, v) = 0

for v E V and t - tj. It follows that

(Ote(tj), v) + At3 (e(tj), v) = (Tj, v) ,

(3.123)

3.5. Alternative Approach

141

where

1 f~~, 1 (~_ "~j = Otu(tj) - ut(tj) = -~

tj_,)~.(~) d~

(3.124)

Putting ~(ty) = tje(tj) and ~j = ty3'j, we get

(Ote(tj), v) 4- At3 (e(tj), v) = (~j 4- e(tj-1), v) . Letting v = ~(tj) now gives that

1 (~(tj) - e(tj-1), ~(tj)) 4- Atj (~(tj), ~(tj)) = (~j 4- e(tj_l), ~(tj)). T In other words, the inequality 1

:2 (ll~-(tj)ll~-II~(tj

-

12) 4- TAt, (~(tj), ~(tj)) = T (X/j 4-

- 1)112 x + r 2 IIO~(tj)

e ( t j _ l ) , e(tj))

holds and we obtain ila(tj)ll ~x - I I ~ ( t j - 1 ) l l

~ + 2 w ~ l ~ ( t j ) l l ~V X

<_ 27 (Sj 4- e(tj_l), ~(tj)) < 2~- { c (llgjll ~v . + I l e ( t j - 1 ) l l v ~. )

+ ~

-

Ila(tj)llv}

9

We have II~(tj)ll X ~ - I l e ( t j - 1 ) l l x<- 2Cw ([l@jl[~. + Ile(tj-1)[2v.)

and hence

t 2 Ile(t,)llx = II~(t,)lI2x <__2 c T

tj 1l'TjIlv. + j=l

Ile(tj)lIv 9

{3.125)

j=l

follows. Equality (3.124) implies ii,),jll2 _~ 1 T

/t;

' (8 - t j _ l ) 2 Ilutt(8)ll 2. d8 1

and hence tj II~jll..

follows because t y ( s with f = 0 implies

_

ftj 'j--1

..

tj-1) <_ Ts for s E [tj-1, tj]. On the other hand, equality (3.110) IIuttllv. < Cllutllv + CIlullv.

(3.126)

3. Evolution Equations and F E M

142

Therefore, we get

n

2

2

< ~-~

j=l

j~0tn

-< c wi

~ I1~.(~)111 V* d~

/o ' (ll~(~)ll~+s ~ll~r

_ cw 2 Ilu011~

by Propositions 3.5 and 3.8. Inequality (3.122) is now a consequence of (3.125). For the operator T(t) =_ A(t) -1, the relation Ilfllv.

~

IIZ(t)fllv

(f ~ Vh)

holds uniformly in h. In use of T'(t) = - A ( t ) - l A ' ( t ) A ( t ) - l ,

I Z'(t)fllv <_ C

Ilfllv.

we have

9

In terms of T(t), equality (3.123) is written as

T(tj)Ote(tj) + e(tj) = T(tj)3,j. Setting Fj = T(tj)~,j, we get

Ot [T(tj)e(tj)] + e(tj) = Fj + [OtT(tj)] e(tj_l). Here we take L 2 inner product with T(tj)e(tj).

We have

(T(tj)e(tj), e(tj)) ~ I e(tj)l

~.

and

(Ot [T(tj)e(ty)] , T(tj)e(ty)) 1 = 2T (llT(tJ)e(tJ) 12X - I l T ( t j - l e ( t j - 1 ) I x

+ 7-IIOt [T(tj)e(tj)]llx),

as in the proof of L e m m a 3.14. It follows that

_1 2

(llZ(ty)e(tj)ll 2x --[IT(tj -1 )e(tj_ 1 )1 2x + ~ II0, (T(tj)e(tj))ll 2 ~) 2 x / + ~ I~(tj)llv. 7<_ Ilrjllv. IIT(tj)e(tj)llv + II[O~T(tj)] e(t~_l) x IIT(tj)e(tj)l x

with a constant # > 0. The right-hand side is estimated from above with some t, 6_ (tj_l, tj)"

I FjlIv. IIT(tj)e(tj)llv § IIT'(t,)e(tj-1)llv T ( t j ) e ( t j ) I x _< c (IIFjlIv. e(tj)llv. + Ile(tj-,)l v. IIZ(tj)e(tj)l x) 1

2

2

_< -~. (lle(tj)llv. + I e(tj_~)llv.) + C (llT(tj)e(tj)ll 2X + IIFj I~*).

(3.127)

Commentary to Chapter 3

143

We obtain 1

9

3

1

2

2

2

+ ~ - , (If~(tj)lf v. -II~(t~- ~)11~.) _< c~- (llT(tj)~(t~)fl~ + Ilrjlt~.) and hence

]]T(tn)e(tn)]] x2 + ~ s

II~(tj)ll~. -< c ~ s

j=l

Ilrjll ~v.

n + c~- ~lf(tj)e(tj)ll2x

j=l

j=l

follows. Then the discrete Gronwall's inequality implies that n

IIT(t~)e(t~)llx + f

Ile(tj)ll V* 2 <- - CT ~ j=l

Ilrjll 2V*

"

(3.128)

j=l

Here, Schwarz's inequality gives JIFjll2v.

IIZ(tj)'~jll~. < 1__

-

T

~ (s - tj_~) 2 ]lZ(tj)u~(s)l[~. ds 1

similarly to the proof of Lemma 3.14. On the other hand, T(t)ut + u = 0 implies

T(t)utt = - u t - T'(t)ut. Therefore, given s E [tj-1, tj] there exists s. E (tj-1, tj) satisfying

<_ Ily(s)utt(s)llv. + f IlY'(s,)u,(s)lIv. _< Ilu~(s)llg. + IIT'(s)~,(s)ltv. + w IlT'(s,)utt(s)llv.

IIT(tj)u,(s)llv.

<__ c (ll~(~)llv. + II~,(~)llv + ~ II~,(~)llv.). In use of [lutllv. = I A(t)ullv.

< C I1~11~, w~ h~v~

IIT(t~)~.(~)llv. <

c

(ll~(*)llv

+ ~

II~,(*)llv)

by (3.126). Consequently, the inequality

~ - ~ Ilrjll ~. _< C j-1

j=l

-1

(~- tj_~ (11~(~)11~+

-<

c~ ~

<

c w 2 IIPu0112

--

(ll~(s)ll =v +

I1~(~)11~)d~

~ Ilu,(~)ll~)ds (3.129)

X

follows from Propositions 3.5 and 3.8. Combining (3.122) with (3.128) and (3.129) gives inequality (3.82):

II~(t)lt~

< CTt-1

II~ollx.

3. Evolution Equations and FEM

144 Commentary

to C h a p t e r 3

3.1. The generation theory of Tanabe-Sobolevskii was given by Tanabe [377] and Sobolevskii [353], independently. Inequality (3.17) is a consequence of the elliptic estimate of Agmon, Douglis, and Nirenberg [3]. The generation theory of Fujie-Tanabe was given by Fujie and Tanabe [132]. Inequality (3.23) follows from the coerciveness of At. The righthand side may be replaced by C~ Izl, but this form simplifies the treatment of fractional powers of A(t). The proof of (3.23) is also given in Tanabe [378]. Generation theory of Kato-Sobolevskii was given by Kato [201] when p-1 is an integer and Sobolevskii [354] for the general case. Inequality (3.27) was made use of by Heinz [169] and Kato [202], [200]. For the theories of fractional powers of m-sectorial operators, see Kato [199]. Inequality (3.26) was also proven there. Equality (3.3o) is due to P.E. Sobolevskii. The theory of Kato-Tanabe was given by Kato and Tanabe [209]. See also Suzuki [368] and ~3.3 for the fact that smoothness of bilinear forms confirm the assumptions of the last theory. Most of those generation theories are described in Tanabe [378] in details. 3.2. Inequality (3.55) was proven by Suzuki [368]. Proof of (3.64) was also given there. Note that necessary assumptions for A(t) are those on the theory of Kato-Tanabe and inequalities (3.60)-(3.62). Once they are provided, the Banach space structure induces the conclusion. 3.3. Inequality (3.67) was proven by Fujita and Suzuki [147] for boundary condition (3.a) and Suzuki [3661, [36s] for the general case. Related works were done by Helfrich [171], Fujita [140], Suzuki [365], and Sammon [335]. 3.4. For the detailed proof of (3.86), see Suzuki [368]. Inequality (3.82) was proven by Suzuki [366], [3681 Related works were done by Sammon [335]. 3.5. Error analysis on the fully discrete approximation of temporally inhomogeneous parabolic equations has been done independently by Baiocchi and Brezzi [23], Huang and Thom~e [184], [185], Luskin and Rannacher [256], [257], and Sammon [336], [337] Contrarily to Suzuki [366], [368], the Hilbert space structure of the problem were used systematically there. The works described in this section were done by Luskin and Rannacher [256] and Huang and ThomSe [185]. The error analysis for schemes with higher accuracy were done by Baiocchi and Brezzi [23], Luskin and Rannacher [257], and Sammon [336], [337].