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Some Aspects of Numerical Solution of Evolution Equations by the Method of Discretization in Time
Numerical Approximationof Partial Differential Equations E.L. Ortiz (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1987
259
SOME ASPECTS OF NUMERICAL SOLUTION OF EVOLUTION EQUATION5 BY THE METHOD OF DISCRETIZATION IN TIME
K a h e l Rektotryn
T e c h n i c a l U n i v e h b i t y Pnague P.tague, C z echonlo v a k i n
Thc method 0 6 d i n c n e t i z a t i u n i n t i m e ( t h e Rothe met h o d , o h h o h i z o n t a l method 0 6 l i n e n ) han been ohown t o be a powetrdul t o o l 0 6 b o l u t i a n ad a bhoad h C a $ e 0 6 e v o l u t i o n phoblemo, a d concetrnn b o t h t h e o h e t i c a l and numetrical quebZ i o n s . The t e c h n i c s , nuggebted b y t h e aut hoh ( R e k t o h y b , 1 9 7 1 / , made it ponni bee t u t h e a t t h e b e c j u e h t i o n b i n a p a k t i c u l a h l y n i m p l e w a y , w h i l e , a t t h e bame t i m e , new h e h u t t b hav e been o b t a i n e d . Some 0 6 t h c m , i n t e h e b t i n g e n p e c i a l l y 6hom t h e p o i n t v 6 v i e w 0 6 numehical a n a l y d i b (convehgence t h e o hemo, i n c l u d i n g convehgence 0 6 t h e “ R i k z - R o t h e method”, ehhoh e n t i m a t e o , e t c . ), a t e ptrebented below. !,lany years ago, E. Rothe suggested an approximate
method of solution of a parabolic problem of the second order in two dimensions. Let us show his idea on the following trivial example: dU
a
ax2
at
=
o
in
,
U(X,O) = U,(X) u(O,t)
( ~ , T I,
(0,~) x
= U ( l , t )
.
= 0
Let us divide the interval intervals
I. 7
’
j = I , . . . , p , of the lengh
points of division
t
j
. For
every
mate the (not yet known) function
z . ( x ) and the derivative 3
quotient
I = [O,Tj
au/at
t
t. 3
into
p
sub-
h = T/p , with let u s approxi-
u(x,t.) by a function 7
in ( 1 ) by the difference
K. Rektorys
260
z.(x) =
Z'(X) 7
- zj-l (x) h
1
Starting by the function zl(x),
...,zp(x)
z
0
(x) = uo(x) , the functions
are determined subsequently as solutions of
ordinary boundary value problems z1
-
zo
+
2"
22
-
z1
+
2"
h h
1
2
= 0
,
= 0
,
z1 ( 0 ) = 0
,
z1 ( L ) = 0
z 2 (0) = 0
,
z2(&) = 0 ,
Having obtained the functions led R o t h e function (0,e) x
I.
7 '
every
x
t = t. 3
,...,z P (x) , the
zl(x)
so-cal-
is defined in the whole region
(O,T) by u1 (x,t)
in
ul(x,t)
,
= Zj-I (XI
j = I,..
.,p
+
'
z . (XI
-
Z.-I
h
)
z
. (x) for every
7
refining the original division, thus construc-
ting divisions with the steps a sequence
1-1
, thus piecewise linear in t for
fixed and assuming the values
. By
(x) (t - t .
{un(x,t) }
h/2, h/4, h/8,
..., we
obtain
of corresponding Rothe functions
which can be expected to converge (in an appropriate space) to the solution
u
(in an appropriate sence) of the given
problem. The problem solved by E. Rothe was a simple one, and for years, the method was forgotten. It was revived by 0.
A.
Ladyienskaja, and further applied by other renowned mathematicians to the solution of substantially more complicated problems (parabolic problems, linear and quasilinear, in
N
dimensions, then problems of arbitrary order, nonlinear problems, the Stephan problem, hyperbolic problems, mixed para-
Method of Discretization in Time
bolic-hyperbolic problems, etc.). Many of the obtained results were obtained, as well, by other methods (method of semigroups, method of monotone oprators, the Fourier method, etc.).
As concerns numerical point of view, related methods
(methods of space-, or time-space discretization) were applied as well. Each of these methods, including the method of time-discretization (= the Rothe method) has its preferences and its drawbacks. However, the Rothe method has its significance both as a numerical method and theoretical tool. Existence theorems are proved in a constructive way. Thus no other methods are needed to give preliminary information on existence, or regularity of the solution as required in many other numerical methods when questions on convergence, or order of convergence, etc. are to be answered. The Rothe method is a stable method. To the solution of elliptic problems generated by this method, current methods, especially the variational ones, can be applied. As concerns theoretical results, they are obtained in a relatively simple way, as usual. Moreover, the Rothe method, being a very natural one, makes it possible to get a particularly good insight into the structure of the solutions. This is why I prefer it. In one of my works (Rektorys, 1 9 7 1 ) a slightly different technics was given than that applied currently in this method, making it possible to treat corresponding elliptic problems in a particularly simple way. This technics was followed by other authors and became a base for the work of my seminar at the Technical University in Prague. Results obtained in this seminar were summarized in a monograph
261
K. Rektorys
262
(Rektorys, 1982). I would like to present some of them here, namely those, interesting from the numerical point of view, pointing out the very simple way in which they have been obtained.
1. A c o n v e r g e n c e t h e o r e m . To begin with, consider a
relatively simple parabolic problem
U(X,O) = 0 ,
B.U = 0
on
r
x
( 0 , ~ ),
i = 1
,...,P
0
on
r x
(0,T) ,
i
,...,k-u .
C.U
Here,
=
r , A
dary
A = with
EN
is a bounded region in
G
z
with a Lipschitz boun-
(-l)IilDi(aij (x)DJ) ,
bounded and measurable in
13
(4)
is a linear operator of order 2k, of the form
lil, I j l l c
a. .
1
=
,
(5)
G ,
f E L2 ( G ) ; ( 3 ) ,
or (4) are (linear) boundary conditions, stable (thus containing derivatives of orders respect to the operator
2 k -
or unstable with
1 ),
A , respectively. Denote
V = {v; v E W i k ) (G) , Biv
of traces, i = 1,
=
0
on
r
in the sence
...,u } .
LettheRothe method be applied, thus let, f o r each the derivative
z . (x)
=
au/at
(. 6 ).
t = t. 7 ’
be replaced by
Zi(X) - zi-, (x) h
In the weak formulation, we thus have to find, subsequently, such functions z . f V 3
,
j = I , ...,p
,
(7)
Method of Discretization in Time
263
for which the integral identities ((v,z.)) 3
+
3 v
(V,Z.)= (v,f) 3
are satisfied, with
v
(8)
zo(x) = u(x,O) : 0
the scalar product in
((.,.I)
,
L2(G)
form corresponding to the operator
A
. Here,
is
(.,.)
is the bilinear and to the boundary
conditions ( 3 ) , ( 4 1 , familiar from the theory of variational methods, see, e.g.
-
A
, u
= 0
(Rektorys, 1979). (For example, for
A =
r , we have
on
,
K
Let positive numbers
(independent of
a
v
and
u )
exist such that the inequalities
I ((VSU))I
2 K l Ivl Iv1 IUI Iv *
((v,v)) 2 a l vl hold for all the form
( (.
Iv2
(10)
v, u E V
,.)
(9)
.)
.
(V-boundedness and V-ellipticity of
Under the assumptions (91, (10) each of
the problems ( 7 ) , (8) is uniquely solvable and thus the above mentioned Rothe function can be constructed. To obtain information about the sequence
{un(x,t)}
of these Rothe
functions, some a p r i o r i estimates are needed. Let us draw the attention to the fact in what a simple way they are obtained: Put for
j = 1
v =
. We
Z1 =
(2,
-
zo)/h
=
zl/h into (8) written
get
h((Zl,Z1)) + (Zl,Zl) Because of (10) and
llzlll2 ~ l l Z , I l
I (Z IIf
Subtracting (8), written for
j
- 1 ,
from (81, gives
N
'4
2
I1
u.
Y
2
3
T
I
rt
..A
rt u- 3
I
u- 3
N
I
U.
N
-+
N
u- 3
I1
Y.
. .
Y
a C ( D ( D a
U.
n
% %
a
art
P,
c
O D r t 3
P,
W
IlA n
r t
n
3
c
tk
0 Y
m
Lr
v
4
0 N
II
e n
2
m 0 r Y
rt
r.
?2
N
I
-3
N
+
+
--L v
(D
E
m
a
r u g
r P,r
luN3
II
N
u. 3
rt
7
\ N
g
19
3
0
TJ
0
a
z.
a (D
r.
3
0
n
3
:
2.
r.
a
Y
c
0
5'0
I
tr 3 (D
tr
II
r 2
N
2
II
3
2
m
RY
n
(D
Y
E
rt
(D
r
$ (D
(D
0 P,
0
3
m
v
N
4
-
Y
(D
5
(D
rt 0 9
m
9
5'
rt
ke, N
-
m
N
II
U.
-
U.
(D
Y
M 0
(D
b-
m
P,
9
Lc
"
&
P,
r
t I
UN
I
N
u.
0
tr
"
<
+ h
3
N
u.
3
P
Q)
N
Method of Discretization in Time
265
n = 1,2,.
..
(the Rothe functions), or
n = 1,2,.
..
be abstract functions, considered as functions
I
from
=
[o,TJ
into
v ,
L ~ ( G ), respectively. In con-
or
sequence of their form and of ( 1 6 ) and ( 1 4 ) , they are uniformly bouded (with respect to
L2(I,V) , or
n ) in
L2(I,L2(G))
, respectively (even in C(1,V) , or
L_(I,L2(G))
. The
spaces
L2(I,V) , and
found such that u
nk
A
u
in
L2(I,V)
, or
{un } k
Hilbert spaces, a subsequence
, Un
k
A
U
L2(I,L2(G))
in
{unk'
being
can be
L2(I,L2(G))
.
(19)
It easily follows (Rektorys, 1 9 8 2 , Chap. 11) that
T((v,u)) dt
+
T
(v,u') dt =
A function with the properties ( 2 0 ) w e a k s o l u t . i - o n of the problem ( 1 )
-
-
( 2 3 ) is called the
(4).
Uniqueness is then
proved implying convergence of the whole sequence
{ un)
to
u: Theorem 1 .
L2(G)
.
Let ( 9 ) ,
(10) be satisfied, let
f E
Then there exists exactly one weak solution of the
K. Rektorys
266
-
problem (1) u n
1
u
( 4 ) and
in
L~(I,v) ,
R e m a r k 1.
un
->
u
in
L2(I,L2(G))
.
(24)
By a more detailed treatment it can be
=>
u in C(1,V) , u' E Lm(I,L2(G)) , u n further that the assumption (10) can be weakened, etc. We do
proved that even
not go into details here. See (Rektorys, 1982, Chap. 21).
As can be expected, to get an
2. An e r r o r E s t i m a t e .
efficient error estimate. some supplementary assumptions are needed: Let the assumptions of Theorem 1 be fulfilled. Let, moreover, f E V
, Af E L2(G) ,
v v E
((v,f)) = (v,Af)
V.
(25
Then
1
ju(x,t.) 7
where
M =
-
Zj(X) 1
1
2
2
Mjh
,
,
j = l,...,~
(26
I / A f // .
Using the technics applied above, the estimate (26) is easily derived, estimating, for
rence between
z.(x) 3
t
=
t. 3
fixed, the diffe-
and corresponding functions, obtained
by the Rothe method when refining the division of the interval
I
and coming to the limit (Rektorys, 1982, Chap. 1 2 ) .
For these estimations, uniform boundedness, in
L2(G) , of
higher difference quotients is needed, what is ensured when (25) is fulfilled.
From among the assumptions (25), only the first is restrictive in applications. The estimate (26) is very sharp, as seen from a lot of numerical examples given in (Rektorys, 19821, and cannot be improved substantially.
267
Method of Discretization in Time
3
C o n v e r g e n c e of t h e R i t z - R o t h e
method.
Let the
7), ( 8 ) be solved approximately, to be concrete,
problems
by the Ritz method (or by a method with similar properties).
...
We speak briefly of the "Ritz-Rothe method. So let
...,vn
be the first
n
terms of a base in
be the Ritz approximation of the function stead of
z1
and let
. Put
z;
in-
into the second of the identities ( 8 ) ,
z1
and let
V
vl,
be the Ritz approximation of the function
%
z2 ,
etc. We thus can construct the function
*
*
+
= z;-l
u;w
for
(t t L t
tj-l
-
tj-,)
,
j = l,...,p
j '
which is an analogue of the Rothe function
ul(t)
.
(27)
announces that using the Ritz method, the errors become cumulated with increasing
j
e
However, according to a very
simple law: Substract (27) from the second of the identities (8). We obtain
((v,
- 22)) +
2*
1
L(V, ( z 2
-
%
2,)
-
(zl
-
2;))
= 0
V V E V .
v
Putting
= z
- z2 , %
we get
Etc. Using this result, in (Rektorys, 1982, Chap. 14) the convergence of the Ritz-Rothe method is then proved in the following sense:
1
Iu
-
* u1
6
being given arbitrarily,
> 0
I IC (I,L2 ( G ) )
can be achieved if
h
<
E
is sufficiently small and the number
K. Rektorys
268
of basic elements in the Ritz method is sufficiently large. Remark 2.
type 1 , 2 , 3
Basic ideas how to obtain results of the
were shown on the rather simple example ( 1 )
with time-independent operator
A
-
(4)
and homogeneous initial
and boundary conditions. In (Rektorys, 1 9 8 2 1 , the case of nonhomogeneous initial and boundary conditions is discussed, then the case
A = A(t) ,
A
ferential parabolic problems
nonlinear, further integrodifare treated, problems with an
integral condition, etc. Existence and convergence theorems (including convergence of the Ritz-Rothe method) are proved. Hyperbolic problems are then treated in a rather nontraditional way and an error estimate, similar to the estimate ( 2 6 1 , is derived. See also Pultar, 1 9 8 4 . Remark 3.
In (Rektorys, 1 9 8 2 ) , the technics shown
above is applied as well to obtain results of purely theoretical character, for example regularity results. In particular, the "smoothing effect" for a homogeneous parabolic equation with a nonhomogeneous initial condition uo E L 2 ( G )
u(x,O) =
is derived in a relatively very simple way.
REFERENCES 1. Rektorys, K . , On Application of Direct Variational Methods to the Solution of Parabolic Boundary Value Problems of Arbitrary Order. Czech. Math. J. 21, 1 9 7 1 , 318-339. 2. Rektorys, K., Variational Methods in Mathematics, Science and Engineering, 2nd Ed., D. Reidel, Dordrecht-BostonLondon, 1979. 3 . Rektorys, K., The Method of Discretization in Time and Partial Differential Equations, D. Reidel, DordrechtBoston-London, 1982. 4. Pultar, M., Solution of Abstract Hyperbolic Equations by Rothe Method. Aplikace matematiky 29, 1 9 8 4 , 2 3 - 3 9 .
259
SOME ASPECTS OF NUMERICAL SOLUTION OF EVOLUTION EQUATION5 BY THE METHOD OF DISCRETIZATION IN TIME
K a h e l Rektotryn
T e c h n i c a l U n i v e h b i t y Pnague P.tague, C z echonlo v a k i n
Thc method 0 6 d i n c n e t i z a t i u n i n t i m e ( t h e Rothe met h o d , o h h o h i z o n t a l method 0 6 l i n e n ) han been ohown t o be a powetrdul t o o l 0 6 b o l u t i a n ad a bhoad h C a $ e 0 6 e v o l u t i o n phoblemo, a d concetrnn b o t h t h e o h e t i c a l and numetrical quebZ i o n s . The t e c h n i c s , nuggebted b y t h e aut hoh ( R e k t o h y b , 1 9 7 1 / , made it ponni bee t u t h e a t t h e b e c j u e h t i o n b i n a p a k t i c u l a h l y n i m p l e w a y , w h i l e , a t t h e bame t i m e , new h e h u t t b hav e been o b t a i n e d . Some 0 6 t h c m , i n t e h e b t i n g e n p e c i a l l y 6hom t h e p o i n t v 6 v i e w 0 6 numehical a n a l y d i b (convehgence t h e o hemo, i n c l u d i n g convehgence 0 6 t h e “ R i k z - R o t h e method”, ehhoh e n t i m a t e o , e t c . ), a t e ptrebented below. !,lany years ago, E. Rothe suggested an approximate
method of solution of a parabolic problem of the second order in two dimensions. Let us show his idea on the following trivial example: dU
a
ax2
at
=
o
in
,
U(X,O) = U,(X) u(O,t)
( ~ , T I,
(0,~) x
= U ( l , t )
.
= 0
Let us divide the interval intervals
I. 7
’
j = I , . . . , p , of the lengh
points of division
t
j
. For
every
mate the (not yet known) function
z . ( x ) and the derivative 3
quotient
I = [O,Tj
au/at
t
t. 3
into
p
sub-
h = T/p , with let u s approxi-
u(x,t.) by a function 7
in ( 1 ) by the difference
K. Rektorys
260
z.(x) =
Z'(X) 7
- zj-l (x) h
1
Starting by the function zl(x),
...,zp(x)
z
0
(x) = uo(x) , the functions
are determined subsequently as solutions of
ordinary boundary value problems z1
-
zo
+
2"
22
-
z1
+
2"
h h
1
2
= 0
,
= 0
,
z1 ( 0 ) = 0
,
z1 ( L ) = 0
z 2 (0) = 0
,
z2(&) = 0 ,
Having obtained the functions led R o t h e function (0,e) x
I.
7 '
every
x
t = t. 3
,...,z P (x) , the
zl(x)
so-cal-
is defined in the whole region
(O,T) by u1 (x,t)
in
ul(x,t)
,
= Zj-I (XI
j = I,..
.,p
+
'
z . (XI
-
Z.-I
h
)
z
. (x) for every
7
refining the original division, thus construc-
ting divisions with the steps a sequence
1-1
, thus piecewise linear in t for
fixed and assuming the values
. By
(x) (t - t .
{un(x,t) }
h/2, h/4, h/8,
..., we
obtain
of corresponding Rothe functions
which can be expected to converge (in an appropriate space) to the solution
u
(in an appropriate sence) of the given
problem. The problem solved by E. Rothe was a simple one, and for years, the method was forgotten. It was revived by 0.
A.
Ladyienskaja, and further applied by other renowned mathematicians to the solution of substantially more complicated problems (parabolic problems, linear and quasilinear, in
N
dimensions, then problems of arbitrary order, nonlinear problems, the Stephan problem, hyperbolic problems, mixed para-
Method of Discretization in Time
bolic-hyperbolic problems, etc.). Many of the obtained results were obtained, as well, by other methods (method of semigroups, method of monotone oprators, the Fourier method, etc.).
As concerns numerical point of view, related methods
(methods of space-, or time-space discretization) were applied as well. Each of these methods, including the method of time-discretization (= the Rothe method) has its preferences and its drawbacks. However, the Rothe method has its significance both as a numerical method and theoretical tool. Existence theorems are proved in a constructive way. Thus no other methods are needed to give preliminary information on existence, or regularity of the solution as required in many other numerical methods when questions on convergence, or order of convergence, etc. are to be answered. The Rothe method is a stable method. To the solution of elliptic problems generated by this method, current methods, especially the variational ones, can be applied. As concerns theoretical results, they are obtained in a relatively simple way, as usual. Moreover, the Rothe method, being a very natural one, makes it possible to get a particularly good insight into the structure of the solutions. This is why I prefer it. In one of my works (Rektorys, 1 9 7 1 ) a slightly different technics was given than that applied currently in this method, making it possible to treat corresponding elliptic problems in a particularly simple way. This technics was followed by other authors and became a base for the work of my seminar at the Technical University in Prague. Results obtained in this seminar were summarized in a monograph
261
K. Rektorys
262
(Rektorys, 1982). I would like to present some of them here, namely those, interesting from the numerical point of view, pointing out the very simple way in which they have been obtained.
1. A c o n v e r g e n c e t h e o r e m . To begin with, consider a
relatively simple parabolic problem
U(X,O) = 0 ,
B.U = 0
on
r
x
( 0 , ~ ),
i = 1
,...,P
0
on
r x
(0,T) ,
i
,...,k-u .
C.U
Here,
=
r , A
dary
A = with
EN
is a bounded region in
G
z
with a Lipschitz boun-
(-l)IilDi(aij (x)DJ) ,
bounded and measurable in
13
(4)
is a linear operator of order 2k, of the form
lil, I j l l c
a. .
1
=
,
(5)
G ,
f E L2 ( G ) ; ( 3 ) ,
or (4) are (linear) boundary conditions, stable (thus containing derivatives of orders respect to the operator
2 k -
or unstable with
1 ),
A , respectively. Denote
V = {v; v E W i k ) (G) , Biv
of traces, i = 1,
=
0
on
r
in the sence
...,u } .
LettheRothe method be applied, thus let, f o r each the derivative
z . (x)
=
au/at
(. 6 ).
t = t. 7 ’
be replaced by
Zi(X) - zi-, (x) h
In the weak formulation, we thus have to find, subsequently, such functions z . f V 3
,
j = I , ...,p
,
(7)
Method of Discretization in Time
263
for which the integral identities ((v,z.)) 3
+
3 v
(V,Z.)= (v,f) 3
are satisfied, with
v
(8)
zo(x) = u(x,O) : 0
the scalar product in
((.,.I)
,
L2(G)
form corresponding to the operator
A
. Here,
is
(.,.)
is the bilinear and to the boundary
conditions ( 3 ) , ( 4 1 , familiar from the theory of variational methods, see, e.g.
-
A
, u
= 0
(Rektorys, 1979). (For example, for
A =
r , we have
on
,
K
Let positive numbers
(independent of
a
v
and
u )
exist such that the inequalities
I ((VSU))I
2 K l Ivl Iv1 IUI Iv *
((v,v)) 2 a l vl hold for all the form
( (.
Iv2
(10)
v, u E V
,.)
(9)
.)
.
(V-boundedness and V-ellipticity of
Under the assumptions (91, (10) each of
the problems ( 7 ) , (8) is uniquely solvable and thus the above mentioned Rothe function can be constructed. To obtain information about the sequence
{un(x,t)}
of these Rothe
functions, some a p r i o r i estimates are needed. Let us draw the attention to the fact in what a simple way they are obtained: Put for
j = 1
v =
. We
Z1 =
(2,
-
zo)/h
=
zl/h into (8) written
get
h((Zl,Z1)) + (Zl,Zl) Because of (10) and
llzlll2 ~ l l Z , I l
I (Z IIf
Subtracting (8), written for
j
- 1 ,
from (81, gives
N
'4
2
I1
u.
Y
2
3
T
I
rt
..A
rt u- 3
I
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Method of Discretization in Time
265
n = 1,2,.
..
(the Rothe functions), or
n = 1,2,.
..
be abstract functions, considered as functions
I
from
=
[o,TJ
into
v ,
L ~ ( G ), respectively. In con-
or
sequence of their form and of ( 1 6 ) and ( 1 4 ) , they are uniformly bouded (with respect to
L2(I,V) , or
n ) in
L2(I,L2(G))
, respectively (even in C(1,V) , or
L_(I,L2(G))
. The
spaces
L2(I,V) , and
found such that u
nk
A
u
in
L2(I,V)
, or
{un } k
Hilbert spaces, a subsequence
, Un
k
A
U
L2(I,L2(G))
in
{unk'
being
can be
L2(I,L2(G))
.
(19)
It easily follows (Rektorys, 1 9 8 2 , Chap. 11) that
T((v,u)) dt
+
T
(v,u') dt =
A function with the properties ( 2 0 ) w e a k s o l u t . i - o n of the problem ( 1 )
-
-
( 2 3 ) is called the
(4).
Uniqueness is then
proved implying convergence of the whole sequence
{ un)
to
u: Theorem 1 .
L2(G)
.
Let ( 9 ) ,
(10) be satisfied, let
f E
Then there exists exactly one weak solution of the
K. Rektorys
266
-
problem (1) u n
1
u
( 4 ) and
in
L~(I,v) ,
R e m a r k 1.
un
->
u
in
L2(I,L2(G))
.
(24)
By a more detailed treatment it can be
=>
u in C(1,V) , u' E Lm(I,L2(G)) , u n further that the assumption (10) can be weakened, etc. We do
proved that even
not go into details here. See (Rektorys, 1982, Chap. 21).
As can be expected, to get an
2. An e r r o r E s t i m a t e .
efficient error estimate. some supplementary assumptions are needed: Let the assumptions of Theorem 1 be fulfilled. Let, moreover, f E V
, Af E L2(G) ,
v v E
((v,f)) = (v,Af)
V.
(25
Then
1
ju(x,t.) 7
where
M =
-
Zj(X) 1
1
2
2
Mjh
,
,
j = l,...,~
(26
I / A f // .
Using the technics applied above, the estimate (26) is easily derived, estimating, for
rence between
z.(x) 3
t
=
t. 3
fixed, the diffe-
and corresponding functions, obtained
by the Rothe method when refining the division of the interval
I
and coming to the limit (Rektorys, 1982, Chap. 1 2 ) .
For these estimations, uniform boundedness, in
L2(G) , of
higher difference quotients is needed, what is ensured when (25) is fulfilled.
From among the assumptions (25), only the first is restrictive in applications. The estimate (26) is very sharp, as seen from a lot of numerical examples given in (Rektorys, 19821, and cannot be improved substantially.
267
Method of Discretization in Time
3
C o n v e r g e n c e of t h e R i t z - R o t h e
method.
Let the
7), ( 8 ) be solved approximately, to be concrete,
problems
by the Ritz method (or by a method with similar properties).
...
We speak briefly of the "Ritz-Rothe method. So let
...,vn
be the first
n
terms of a base in
be the Ritz approximation of the function stead of
z1
and let
. Put
z;
in-
into the second of the identities ( 8 ) ,
z1
and let
V
vl,
be the Ritz approximation of the function
%
z2 ,
etc. We thus can construct the function
*
*
+
= z;-l
u;w
for
(t t L t
tj-l
-
tj-,)
,
j = l,...,p
j '
which is an analogue of the Rothe function
ul(t)
.
(27)
announces that using the Ritz method, the errors become cumulated with increasing
j
e
However, according to a very
simple law: Substract (27) from the second of the identities (8). We obtain
((v,
- 22)) +
2*
1
L(V, ( z 2
-
%
2,)
-
(zl
-
2;))
= 0
V V E V .
v
Putting
= z
- z2 , %
we get
Etc. Using this result, in (Rektorys, 1982, Chap. 14) the convergence of the Ritz-Rothe method is then proved in the following sense:
1
Iu
-
* u1
6
being given arbitrarily,
> 0
I IC (I,L2 ( G ) )
can be achieved if
h
<
E
is sufficiently small and the number
K. Rektorys
268
of basic elements in the Ritz method is sufficiently large. Remark 2.
type 1 , 2 , 3
Basic ideas how to obtain results of the
were shown on the rather simple example ( 1 )
with time-independent operator
A
-
(4)
and homogeneous initial
and boundary conditions. In (Rektorys, 1 9 8 2 1 , the case of nonhomogeneous initial and boundary conditions is discussed, then the case
A = A(t) ,
A
ferential parabolic problems
nonlinear, further integrodifare treated, problems with an
integral condition, etc. Existence and convergence theorems (including convergence of the Ritz-Rothe method) are proved. Hyperbolic problems are then treated in a rather nontraditional way and an error estimate, similar to the estimate ( 2 6 1 , is derived. See also Pultar, 1 9 8 4 . Remark 3.
In (Rektorys, 1 9 8 2 ) , the technics shown
above is applied as well to obtain results of purely theoretical character, for example regularity results. In particular, the "smoothing effect" for a homogeneous parabolic equation with a nonhomogeneous initial condition uo E L 2 ( G )
u(x,O) =
is derived in a relatively very simple way.
REFERENCES 1. Rektorys, K . , On Application of Direct Variational Methods to the Solution of Parabolic Boundary Value Problems of Arbitrary Order. Czech. Math. J. 21, 1 9 7 1 , 318-339. 2. Rektorys, K., Variational Methods in Mathematics, Science and Engineering, 2nd Ed., D. Reidel, Dordrecht-BostonLondon, 1979. 3 . Rektorys, K., The Method of Discretization in Time and Partial Differential Equations, D. Reidel, DordrechtBoston-London, 1982. 4. Pultar, M., Solution of Abstract Hyperbolic Equations by Rothe Method. Aplikace matematiky 29, 1 9 8 4 , 2 3 - 3 9 .