O P E R A T O R THEORY AND N U M E R I C A L M E T H O D S
STUDIES IN MATHEMATICS AND ITS APPLICATIONS
VOLUME
30
Editors: Paris PAPANICOLAOU, New York H . F U J I T A , Tokyo H . B . K E L L E R , Pasadena J.L. LIONS,
G.
ELSEVIER A M S T E R D A M - LONDON - NEW YORK - OXFORD - PARIS - S H A N N O N - TOKYO
OPERATOR THEORY AND NUMERICAL METHODS
HIROSHI
FUJITA
The Research Institute of Educational Development Tokai University Tokyo, Japan
NORIKAZU
SAITO
Faculty of Education Toyama University Toyama, Japan and
TAKASHI
SUZUKI
Department of Mathematics, Graduate School of Science Osaka University Toyonaka, Japan
2001 ELSEVIER AMSTERDAM-
LONDON - NEW YORK - OXFORD - PARIS - SHANNON-
TOKYO
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9 2001 Elsevier Science B.V. All rights reserved.
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Preface Recent developments in numerical computations are amazing. A lot of huge projects in applied and theoretical sciences are becoming successful by them, while similar things are happening even in the level of personal computers. Under such a situation, theoretical studies on numerical schemes are fruitful and highly needed. The purpose of the present book is to provide some of them, particularly for schemes to solve partial differential equations. In 1991, we published an article on the finite element method applied to evolutionary problems from Elsevier Publishers (Fujita and Suzuki [148]). This book follows basically that way of description. We study various schemes from the operator theoretical point of view. Many parts are devoted to the finite element method, of which history is described in Oden [306]. We deal with elliptic and then time dependent problems in use of the semigroup theory and so forth. Some other schemes and problems are also discussed, with the later development taken also into account. We are led to believe that any scheme used practically has significant and tight structures mathematically and the converse is also true. Besides some corrections and supplementary descriptions, we have added the following topics: (1) /Y' estimate of the Ritz operator associated with the finite element approximation (2) asymptotic expansions and Richardson's extrapolation for the finite element solution (3) Trotter-Lee's product formula for holomorphic semigroups (4) mixed finite element method (5) Nehari's iterative method for non-stable solutions of elliptic problems (6) finite element approximation of nonlinear semigroups and applications (7) boundary element method for elliptic problems (8) charge simulation method for elliptic problems (9) domain decomposition method for elliptic problems. To make the description consistent, the domain in consideration is mostly supposed to be a convex polygon, and piecewise linear trial functions are adopted unless otherwise stated. The other cases are described in the notes at the end of chapters. The authors thank Professors M. Katsurada, A. Mizutani, H. Okamoto, and T. Tsuchiya for reminding us of some recent developments in the theoretical study and actual computations. Thanks are also due to Ms Y. Ueoka for typesetting the manuscript carefully. Tokyo / Toyama / Osaka April 2001 Hiroshi FUJITA Norikazu SAITO Takashi SuzuKI
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Contents Preface 1
Elliptic Boundary Value Problems and FEM 1.1 Elliptic B o u n d a r y Value P r o b l e m s . . . . . . . 1.2 R i t z - G a l e r k i n M e t h o d . . . . . . . . . . . . . 1.3 F i n i t e E l e m e n t M e t h o d ( F E M ) . . . . . . . . 1.4 Inverse A s s u m p t i o n . . . . . . . . . . . . . . . 1.5 L ~ E s t i m a t e . . . . . . . . . . . . . . . . . . 1.6 L p E s t i m a t e . . . . . . . . . . . . . . . . . . . 1.7
2
v
Asymptotic Expansion
. . . . . .
1 1 7 10 14 19 27
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Semigroup Theory and FEM 2.1 E v o l u t i o n a r y P r o b l e m s . . 2.2 S e m i - d i s c r e t i z a t i o n . . . . 2.3 F r a c t i o n a l Powers . . . . . 2.4 F u l l - d i s c r e t i z a t i o n . . . . . 2.5 I n h o m o g e n e o u s E q u a t i o n . 2.6 H i g h e r A c c u r a c y . . . . . 2.7 L ~ E s t i m a t e . . . . . . . 2.8 H y p e r b o l i c E q u a t i o n . . .
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95 108 118 126 131
4
Other Methods in Time Discretization 4.1 R a t i o n a l A p p r o x i m a t i o n of S e m i g r o u p s . . . . . . . . . . . . . . . . . . . . 4.2 M u l t i - s t e p M e t h o d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 P r o d u c t F o r m u l a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
145 145 153 165
5
Other Methods in Space Discretization 5.1 L u m p i n g of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 U p w i n d F i n i t e E l e m e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . .
171 171 181
3
Evolution 3.1 3.2 3.3 3.4 3.5
Equations
. . . . . . . .
. . . . . .
and FEM
Generation Theories A Priori Estimates . Semi-discretization . Full-discretization . . Alternative Approach
. . . . .
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95
vii
5.3 5.4 5.5
Mixed Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary Element Method (BEM) . . . . . . . . . . . . . . . . . . . . . . Charge Simulation M e t h o d (CSM) . . . . . . . . . . . . . . . . . . . . . .
182 190 195
Nonlinear Problems
207
6.1 6.2 6.3
207 211 217
Semilinear Elliptic E q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . Semilinear P a r a b o l i c E q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . Degenerate Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . .
Domain Decomposition Method 7.1 7.2 7.3 7.4 7.5 7.6
243
Dirichlet to N e u m a n n (DN) Map . . . . . . . . . . . . . . Dirichlet to N e u m a n n (DN) I t e r a t i o n . . . . . . . . . . . . Dirichlet 2 to N e u m a n n 2 ( D D - N N ) I t e r a t i o n . . . . . . . . . . . . . . . . . . Robin to Robin (RR) Iteration . . . . . . . . . . . . . . . . Exterior P r o b l e m . . . . . . . . . . . . . . . . . . . . . . . T h e Stokes S y s t e m . . . . . . . . . . . . . . . . . . . . . .
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243 246 255 257 258 261
Bibliography
275
Index
303
viii
Chapter
Elliptic
1
Boundary
Value
Problems
and
FEM
The present chapter is concerned with the finite element method (FEM) applied to elliptic boundary value problems. For this topic, we have several monographs such as Strang and Fix [359], Ciarlet [83], Raviart and Thomas [326], Johnson [193], Ciarlet and Lions [85], Szab5 and Babuguka [372], and Brenner and Scott [60]. Here, we describe it in the framework of operator theory, picking up approximate operators of the schemes. This way is natural and efficient, particularly in dealing with time dependent problems, because the finite element method is regarded as a discretization of the underlying variational structure.
1.1
Elliptic
Boundary
Value
Problems
To fix the idea, let f~ C R 2 be a convex polygon, and consider the Poisson equation -Au = f
in f~
(1.1)
with Dirichlet condition
u= 0
on Of~.
(1.2)
This boundary value problem has the weak form, described under the following notations. 1~ LP(ft) denotes the set of p-integrable functions for p E [1, oc) and that of essentially bounded functions for p = oc. Unless otherwise stated we assume functions to be real-valued and Banach spaces real. 2 ~ Wm'P(f~) denotes the Sobolev space, the set of measurable functions with their distributional derivatives up to m-th order belonging to LP(ft), where m = 0, 1, 2 , . . . 3 ~ C~(ft) denotes the set of infinitely many times differentiable functions having compact supports contained in f~.
4 ~ Wo'P(f~) denotes the closure of C~(f~) in
wm'p(~).
5 ~ Hm(f~) and Hg(f~) stand for Wm'2(~) and W~'2(f~), respectively. We set Ivlm = I~lm,a - E, 0 satisfying
(1.13)
IIG~llg~(oa) 5 I~11~v - A II'vl %
(v e v)
(1.14)
by (1.7). Here and henceforth, we set X = L2(f~). In the case t h a t b j ( x ) =_ O,
c(x) :> 0,
and
V = H~(f~),
or
bj(x) -- 0,
c(x) > 0,
a(~) k 0,
and
V = Hl(f~),
we can take A = 0 in (1.14). Then, problem (1.3) is uniquely solvable by Lax-Milgram's theorem. In use of the dual space V' of V, boundedness (1.12) of A( , ) implies the welldefinedness of the b o u n d e d linear operator A : V --+ V' through the relation
A(u, v) = (Au, V)v,,v for u, v E V. On the other hand, identifying X' with X by Riesz' representation theorem provides a triple of Hilbert spaces V C X C V' with continuous and dense inclusions. Let
D(A) = {'u E V [ Au E X}. We shall write the restriction of A : V --, V' to D(A) by the same symbol A. It is regarded as an operator in X , and (1.3) is written as an abstract equation in X if f E X = L2(f~): Au = f
1.1. Elliptic Boundary VMue Problems The elliptic regularity is expressed as
D(A) = H2(f~) n H~(~) for boundary condition (1.9) and
+av=O
D(A) = {v 9 H2(f~)
on0t2
(1.15)
for boundary condition (1.10). Those relations hold if 0t2 is sufficiently smooth for instance. We get a strong solution u 9 H2(t2) of (1.8) with (1.9) or (1.10), whenever A = 0 holds in (1.14). The operator A in X arising in such a way from the bilinear form A( , ) on V x V is called m-sectorial. To describe the meaning of this terminology, let us suppose A = 0 in (1.14) for simplicity, and specify the constant C in (1.12) as C1. We can take natural complex extensions of the Banach space X and the operator A. This means that the functions in X are extended to be complex-valued with the inner product
(f, g)= J(n f(x)9(x) dx and Af = Afl + zAf2 if f = fl + zf2 with fl and f2 being real-valued, where z = v/-Z-1. Denote them by the same symbols X and A. Then, inequality (1.12) keeps to hold, while (1.14) is replaced by
R~ .A(v, v)>__ a Ilvl G
(v c V).
A sector is given as
~0 = {~ ~ c I 0 __ I~rg zl ~ 0}, where 0 9 (0, 7r/2). If cos0 = 5/C1, an elementary calculation gives that the numerical range u(A) of A is included in E0, where
~(.A) _= {.A(~, ~)1 ~ ~ V,
II~llx - 1}.
A fundamental property of the numerical range says a(A) C u(A). The relation u(A) C E0 implies that C \ E0 C p(A), where or(A) and p(A) denotes the spectrum and the resolvent set of A, respectively. More precisely, if we take 01 9 (0, 7r/2) and z 9 C in 01 _ a I~11= '
~
V
(v E V)
(1.21)
Given f E V', the problem in consideration is formulated in an abstract manner so as to find u E V satisfying
A(u, v) = ( f , V)v,,v
(1.22)
for any v E V. Then it is uniquely solvable by Lax-Milgram's theorem. Ritz-Galerkin method approximates (1.22) in the following way: Prepare a family of finite dimensional subspaces {Vh}h>0 of V approximating the latter as h ~ 0. Then, we take the problem to find Uh E Vh satisfying
,A(uh, Xh) -- ( f , Xh)v,,v
(1.23)
for any Xh E Vh. Unique solvability of (1.23) follows from the same reasoning as for (1.22). Let Xh be the space Vh equipped with the topology induced from X, and Ph : X ---+Xh the orthogonal projection. The linear operator Ah : X h --+ Xh is defined through
A(uh, Xh) = (A~uh, Xh) for Uh, Xh E Xh. If f E X, problem (1.23) is equivalent to the equation
AhUh = Phf in the finite dimensional space X h. Stability and error estimate of the approximate solution are verified as follows. Let Rh : V --+ Vh be Ritz operator defined through
for ~h E Vh and u E V. Its well-definedness follows from Lax-Milgram's theorem similarly to the unique solvability of (1.23). If u E V is the solution of (1.22), then the approximate solution uh of (1.23) is nothing but Rhu. This gives the relation
R h A - I = Ah~Ph.
(1.24)
1. Elliptic Boundary Value Problems and F E M If .4( , ) is symmetric, the operator Rh " V --+ Vh is nothing but the orthogonal projection with respect to the inner product .4( , ). Therefore, we have
.A(R,,~, R,~u) _ .a(u, u) and
A(R,,~- ~, R,,~- ~) _ A(x,,- u, x , , - u) for any Xh E Vh. Then boundedness (1.20) and strong coerciveness (1.21) o f . 4 ( , ) imply the stability
IIR,,** Iv _ o Ilullv
(1.25)
and the error estimate
IR,~.,,- **llv _< o inf
Xh CVh
IIx,,- **1 v.
(1.26)
Those relations hold even in the general case of `4" 5r .4. In fact, from (1.21) and (1.22) we have ,5 IIR,,ull~
_< ..4(Rh**, R,,~,) = ..4(**, R,,u) _ C I1~,,~11~
9I~ Iv
and 6 IIRhu - ** I =V
0 such t h a t
~(T) > . ~(T) for any T C fh and h > O. Theorem
1.2. If {Th} is regular, the inequality IIv -
~hVlIL~(T) + h
IV(v -
~hV) llZ~ 0. (Actually, given v C H i ( B ) , extend it on Z} = {(x, y) [ Ix + y[ _~ 1} by the even reflection. Then, we see that 7~-1 is taken as the second eigenvalue of - A with the Neumann b o u n d a r y condition on Z}.) This implies also Iv + P[ l,B by ( ~
+ p), 1) :
( ~ ( v + p), 1) : 0
o of triangulations satisfies the inverse assumption if there exists a constant ~'1 > 0 such that p(T) > .~h
1.4. Inverse Assumption
15
for any T E "rh and h > 0. This is equivalent to assuming
Ilxhllc~(z) < c h-2/" Ilxh IL,(T)
(1.45)
for T C ~-h and Xh E Vh with a constant C > 0 independent of h > 0, where p E [1, eel. It also gives
I VxhllL~(r) --< Ch -~ IlxhllL~(r)
(1.46)
and hence
follows. In this section, we show several inequalities derived form this assumption. Theorem
1.3. There exists/3 > 0 such that
(1.47)
IIA~llx.,x~ < ~h -2 if
{Th} satisfies Proof:
the inverse assumption.
In the case that .4( , ) is strongly coercive and symmetric on V x V, we have
Cllxhllv similarly to (1.19). Given Uh, Vh E Vh, we get from (1.46) that
I(Ahun, vh)l
--
A(u, ~) 0. Combining those inequalities with
I1~11~_< I~11-~" I1~11..(~), we conclude that
I1~11.(~) --< C I~1= IIW I.(~)
(1.67)
with ~ = 1 + ( 1 / 2 - 1/p) > 1. Here, we make use of the following lemma. Lemma
1.11.
The inequality
ITI I1~11~(~) ~
3
II~ll.(y)
(x.68)
holds for each T C Th. Proof: Note t h a t ~ >_ 0. There is a vertex P0 E T attaining the m a x i m u m of ~ on T. Let us identify T with the triangle :# in R 3 put on x3 = 0. Then tTlt IITIIILoO(T)indicates the three dimensional volume of the cylinder
On the other hand, because ~7 > 0 on T, the value
11~711L~(T)dominates the volume of
the three dimensional simplex composed of T and P0. The latter is 1/3 of the three dimensional volume of S. This shows (1.68) and the proof is complete. [] For t _> 0, we set
p(t) = Isupp Jh ( R h u - t)l. From the definition of Jh follows ~ ~ p ( t ) dt
~ TCTh
ITI IIr/llLoo(T) 9
1. Elliptic Boundaly Value Problems and FEM
26
Therefore, by (1.67) and Lemma 1.11 we get
Z
~ p(t) dt
(1.71)
for p e [2, oc). Given Xo E ft and ~c >> 1, we introduce the weight function cr = axo,h by ~(x) : (I 9 - xol ~ + ~:~h ~) '/~ A quick observation guarantees [D'cr~ I < C,,,cr "-1;~1
(1.72)
for Ifl[ = 1 , 2 , . . . and 7 E R with Cz,~ > 0 independent of x0 and h. Also, the inverse assumption implies Z 0 ~ T ~ Tit
===~
(7 ~,~ ]1
011
T
(1.73)
and also max T~rh
maxT o" } < C minT cr --
(1.74)
1.6. L p Estimate
29
uniformly in x0. Then, inequality (1.32) gives (1.75) TETh
for/3 e R 2 and v e H~(f~), satisfying
f a-2-~dx
vlT e
~1~+
-< A~ (
/0
He(T) for any T e Th. Finally, N2h2)-1-?/2 dy
@2 + t~2he)-l-~/2r dr ~ h -~
(1.76)
forT> 0 ashl0. Here, we provide an a priori estimate for -Av=f+V.b
with
inf~
u = 0
on cOf~.
(1.77)
If f = 0, we have
Ilvll~,(~) ~ c~ Ilbll~(~)
(1.78)
for p > 2. To see this, we take the dual exponent p' = p~ (p - 1) e (1, 2] of p e [2, ~ ) . Because ~t is convex, v C / 2 ( f i ) implies
W -- (--A) -1 (172P-2V) e W o 'p' (~~)f] W 2'p' (~-~) and
Ilwllw~.p,(,-,) 0 (i = 1, 2, 3) and a0 > O. It holds t h a t Te',-h
~'(r) : -h ~c~C2c3[ 48a~
=
h?
c~c2q, 48a~2
.,,,
D:,D~u. D2v,, dx + C?h'4",'~eT,,']' (T)
D2D3D~u.vhdx+C~h. 4 Z J ' ( T ) . 7"E Th
The terms I2(T) and I3(T) are treated similarly. Letting
Di=Di,
if
i=i'
(mod3),
1.7. Asymptotic Expansion
39
we get 3
I(vh) = h2 ~ D4u "Vh dx + h4 E c4ji' i=1
where D 4 __
3
1
48ao2 E
C4Ci+lCi+2Di+lDi+2D~
i=1
and TCrh
dS.
Oni(T)
i(T)
Therefore, it holds that
(~
- ~)
(~o)
:
h~ s D~
9g~o d~
+h c4Z 3
i=1
T6Th
b~T(S)D4u 9 Og)~ dS.
The first term of the right-hand side of (1.95) is equal to
- A e = D4u
in f~
and
e= 0
(1.95)
O~(T)
i(T)
h2eh with eh = Rhe
for
on 0~.
Here, we have
Ileh ~IIL~(~) --< C h= (llog hi + -
1)IID4UIIL~(a)
by (1.57). To handle with the second term, we suppose xo ~ Fh and recall the function (~o used in the proof of Theorem 1.14. Then, the regularized Green's function {TxoC H2(~2)N H~(f~) is defined by
We have g~ho =
Rh[Txoby
(1.84). In use of ,
E
TCTh
i(T)
OOxo b~r(s)D~uoni(T) dS
=0,
we obtain
E fL rCTh
i(T)
b~(s)D4u 9 Og~~ dS[ Oni(T)
!
TE'rh
Oni(T)
i(T)
_ 0,
~ _> 0,
x~ + ~ < 1}
is indicated as
II/IILI(OB < C (tl/IILI(B)+ t]~7/IILI(B)) In use of the affine mapping ~ : T ---+ B of w
(/E WI'I) .
this gives
Ilfll~(r/- C (h -~ II/IIL*(T/ + I v Z ~l(r/)
(f e W~'~(T))
uniformly in h. We obtain
[Ji I 0,
O'lt ~_>0,
071, u.~-0
onO~.
Variational inequalities can thus formulate free boundary problems. For more details, see Kinderlehrer and Stampacchia [224], Baiocchi and Capelo [35], and Friedman [130].
Commentary to Chapter 1
43
1.2. Inhomogeneous boundary value problem (1.8) with (1.99) is discretized as finding satisfying U h -- UhO E V and .A(Uh, 'Vh) --- (f, Vh) for any Vh C Vh, where Vh C V, ~rh C ~2", and Uh0 9 ~'rh are approximations of V -- H01(gt), 19 = HI(~), and u0 9 V, respectively. The proof of Theorem 1.1 is based on the idea of Agmon [2]. Estimate (1.28) holds for any self-adjoint operator, and the proof is described in standard monographs of functional analysis. Fractional powers A 0 of a non-negative self-adjoint operator A in a Hilbert space is defined through its spectral decomposition. They are extended to m-sectorial operators. Kato [202], [200] showed an inequality of Heinz' type for them. See also Tanabe [378] for this topic. The question D ( A 1/2) = V is referred to as Kato's square root problem. If A is selfadjoint, it holds obviously. T. Kato showed D ( A ~) = D(A~o) for /~ 9 [0, 1/2), where A0 denotes the symmetric part of A described in w J.L. Lions [246] showed that D ( A 1/2) = V actually holds if A is associated with the second order elliptic operator and Hl(f~) C V c Hl(f~). For the general case, however, there are counter examples. See McIntosh [263] for later developments on this problem. Uh E ~rh
1.3. Inequality (1.32) on the interpolation operator 7oh is due to Bramble and Hilbert [45], [46]. For Poincar6-Wirtinger's inequality, see the references to w on real analysis or functional analysis. The interpolation operator 7rh associated with piecewise linear trial functions has the property II~hV --
~llw~,~(a> --< C~ h~-k Ilvl w~,p n/2 (i.e., W2'P(a) C C(~)), and {7h} is regular. Curved elements were introduced by Ciarlet and Raviart [87], ZlSmal [418], [419]. Many works were devoted the theory of approximation such as Yamamoto and Tokuda [409], Ciarlet and Raviart [86], Ciarlet and Wagschal [89], and so forth. For details and later developments, see Ciarlet [83], [84] and the references therein. Some of the earlier studies on the finite element method were Nitsche [296], Bramble and Schatz [55], and Bramble and Nitsche [47]. See also the survey articles Babu~ka and aziz [19] and Bramble [44]. The duality argument is sometimes called Nitsche's trick. Finite element approximation by piecewise linear trial functions for second order elliptic boundary value problems is valid even for three dimensional problems because of the inclusion H2(gt) C C(~). All results presented in this chapter keep to hold in that case, unless otherwise stated. From the proof of Theorem 1.2, we see that the family of interpolation operators {Trh} has the property limh,0 llTChV-- vll V = 0 for v 9 wl,p(f~)ffl v with p > n. Concerning the method employing polynomials of higher degrees as trial functions, we refer to Ciarlet [83], [84] and Brenner and Scott [60]. The method of a posteriori error estimate was proposed by Babu~ka and Rheinboldt [21]. Note that all error estimates presented in this section are a priori. Recently, the method of p- or hp- finite elements by I. Babu~ka is attracting interest, because the development of the hardware makes it clearer that triangulating f~ is not trivial. There, the triangulation is basically fixed and the degrees of trial functions are
1. Elliptic Boundary Value Problems and FEM
44
made higher. See Schwab [346] for details. For the finite element method applied to other problems including variational inequalities, see Galligani and Magenes [156], Glowinski [161], and so forth. Approximation of non-parametric minimal surface was studied by Johnson and Thom6.e [194]. For the parametric case, Tsuchiya [389], [390], [391] showed the convergence of the discrete minimal surfaces constructed by the finite element method. 1.4. Descloux's iemma was shown in [111]. Some generalizations of Propositions 1.5 and 1.7 were given by Crouzeix and Thom~e [102]. Related works are also referred to there. For Riesz-Thorin's interpolation theorem, see Adams [1], Folland [127], and so forth. Other inequalities driven from the inverse assumption are given in Ciarlet [83]. 1.5. The L ~ stability was proven by Nitsche [297] and Scott [349]. R. Scott showed it for - A + 1 under the Neumann boundary condition. Related works were done by Natterer [285], Nitsche [298], [300], Schatz and Wahlbin [345], Rannacher [321] and Finzi Vita [125]. See also the monographs Ciarlet [83], [84], and Brenner and Scott [60]. A related topic is the local estimate of the finite element solution. Several papers were devoted to it: Nitsche and Schatz [302], Bramble and ThomSe [59], Bramble, Nitsche and Schatz [48], Nitsche [299], Bramble and Schatz [56], Schatz and Wahlbin [341], [342], [343], [344], and Thom4e [382]. We also have the survey article Wahlbin [401]. Discrete Green's function was introduced by Frehse and Rannacher [128]. We have also the following result on the optimality of (1.57). Namely, if f~ C R 2 is a disc: f2 = {x I lxl 0
(a E Zh U 13h) ,
A(Wa, Wb) 0
~
u>0 m
and max uj < c max uj
jENo
(1.101)
jEN+
with c = 1, where u = (uj), v = (vj), No = { i l v i = 0}, and N+ = { i l v i > 0}. Matrices satisfying such a condition were studied by Stoyan [358]. If c = 1 is not prescribed, inequality (1.101) is referred to as the general m a x i m u m principle. It is actually equivalent to the general m a x i m u m principle of Schatz if the matrix A is associated with the finite element approximation concerning the inhomogeneous b o u n d a r y condition. The W I'p regularity for the elliptic boundary value problem was shown by Meyers [264]. The W I'p stability of Ritz operator was obtained by Rannacher and Scott [323]. For Sobolev's imbedding theorem (or Sobolev's inequality), see the standard monographs introduced in the references to w Theorem 1.8 was shown by Matsuzawa and Suzuki [262], based on the idea of Suzuki and Fujita [370]. There, the m e t h o d of H a r t m a n and Stampacchia [166] was applied. The result holds also for the N e u m a n n boundary value problem. In the three dimensional case, estimate (1.61) holds with p > 3 if {Th} satisfies the following: any face Zl of any
simplex T E 7h is an acute or right triangle and the line containing the vertex Po (~ T1 of T and perpendicular to T1 passes through T1. As is described at the end of this section, in both cases of two and three space dimensions, inequality (1.59) also holds true with p > n under the same assumptions for {Th}, and therefore, inequality (1.69) follows for q E [1, n / ( n - 1)). There, the convexity of fi is not necessary. Those facts were shown in Matsuzawa [261]. The continuous version of T h e o r e m 1.12 was obtained by Brezis and Strauss [65]. We have
Ilvllwl,~(~/~ Cq IIA~IIL, n: n
A -1
/0+
__ j;;~ L~(fl)
p--0
46
1. Elliptic Boundaly Value Problems and FEM
1.6. Inequality (1.70) follows similarly to Theorem 1.2. The proof is described in standard monographs such as Brenner and Scott [60] and so forth. Even if Rh is replaced by rrh, inequality (1.71) keeps to hold. Furthermore, we see that the order O(h 2) is best possible to control Ilrrhu--Ullg~(n ). Regarding such a situation, we say that (1.71) is
optimal. 1.7'. The asymptotic expansion of the approximate solution was noticed by a series of papers by Q. Lin and his co-workers: Lin and Lu [241], Lin and Tao [242], Lin and Wang [243], Lin and Zhu [245]. See also the survey article Lin and Xie [244]. Relaxations of the uniformity of {rh} were also considered there. Note that if oqf~ is smooth, the elliptic regularity guarantees u E C4'~ from f E C2'~ However, because f~ has corners, it is desirable to reduce the assumption on the smoothness of the solution in Theorem 1.16. For this point, see Blum and Rannacher [40]. For the finite difference operator, there is a close relation between the asymptotic expansion of the approximated solution and the general discrete maximum principle. For this topic, see Wasaw [403] and BShmer [42]. The idea for the proof of Lemma 1.18 is due to Blum, Lin, and Rannacher [39]. Frehse and Rannacher [128] showed (1.96) for the case that 0f~ is smooth with the right-hand side refined as O (llog hi). Asymptotic expansion for the parabolic problem was studied by Helfrich [172].
Chapter 2 Semigroup Theory and FEM
In the following three chapters, we study the discretization of time dependent equations, and the present chapter is devoted to the simplest case. Those problems are described in Thom~e [383], [384], but our analysis is based on the resolvent of operators. Similarly to the continuous case, theory of semigroups on Banach spaces establishes the wellposedness of the discretized problem. Error analysis is also made on the same framework, in use of the complex function theory for the study of approximate operators. It may be worth mentioning now that the semigroup is a family of exponential functions of an operator.
2.1
Evolutionary
Problems
In this section we describe how the abstract theory of semigroups is applied to the continuous case, initial b o u n d a r y value problems for parabolic and hyperbolic equations. Unless otherwise stated, we continue to suppose that f~ C R 2 is a convex polygon. Take the typical initial b o u n d a r y value problem, the heat equation ut - Au -- 0
in gt • (0, cx~)
(2.1)
in t2
(2.2)
with the initial condition ult= o -- Uo(X)
and the boundary condition u = 0
on Or2 x (0, cx:~).
(2.3)
This problem is uniquely solvable if the initial function u0 is not so singular. Even if u0 is not continuous, for instance, if only u0 E L2(gt), the solution u(t, z) is quite smooth for t > 0. This fact is sometimes referred to as the smoothing property, and is reflected by the semigroup of operators which we now start to describe. As is mentioned in the previous chapter, the differentiation s = - A with boundary condition (2.3) is realized as a self-adjoint operator A in X = L2(f~) satisfying
D(A) = H~(a)n H~(a). 47
2. Semigroup Theoly and F E M
48
Then, system (2.1)-(2.3) is formulated as an abstract Cauchy problem for the unknown function u = u(t)" [0, oc)-+ X. Namely, du
dt
+ Au = 0
(t > O)
(2.4)
with
~(o) = ~o.
(2.5)
If we introduce {e -tA }t_>0, the semigroup (of operators) generated by - A , then the solution of (2.4) and (2.5) is simply given by
u(t) = ~-'Auo,
(2.6)
as the general theory of E. Hille and K. Yosida guarantees. However, since the present A admits of a complete orthonormal system of eigenfunctions, we can define e -tA through the eigenfunction expansion in the following way. Let {~}~--0 be the complete orthonormal system of eigenfunctions of A associated with tile eigenvalues {A~}~ n=O, 9 A~,~ =AnOn
( ~ , ~ m ) = S n m = ~1
t0
(n=m)
(~ # ~).
Then any v E X is expanded as oo
(2.7) n=O
for ~n = (v, ~n). The element e-tAr E X may be defined by (3O
e-tAr = E
~ne-t~"~n
(2.8)
(t )_ 0).
n=O
Then e -tA is regarded as a bounded operator oil X. In fact, from (2.8) we have oo
oo
II~-~vll~(~) = ~ 1 9 , ~ - ~ 1 ~ = ~ - ~ n=-0 OO
9hi ~
n----0
E 19,i n--0
In particular, e -tA is a contraction:
t2.9/ We have e-tAcpn = e-ta"y)~ by (2.8). This implies e - t A . e - s A y -- e - ( t + s ) A y
2.1. Evolutionary Problems
49
for v = ~ n and hence for any v E X. On the other hand, e-tA[t=O =--- I (the identity operator) is obvious. Thus the family of operators {U(t)}t>_o has the semigroup property, where U(t) = e-tA:
u ( t ) . u(~)
=
u ( t + ~)
U(O)
=
I.
(t, ~ >_ o)
In view of (2.9), we can say that {e-tA}t>_o is a contraction semigroup. strong continuity of t ~-+ e-tAr, lim lie-tAr -- filL2 = 0 tl0 (n)
(V E X )
Moreover, the
'
is proven. Thus the family {e -tn}t>_o forms a strongly continuous semigroup, referred to as the (Co) semigroup. The condition for v E X of (2.7) to belong to D(A) is oo
19~,, ~ < +o~, n=0
which is equivalent to the convergence of oo
fi/nAn~n
in X.
n=0
In use of the elementary inequality
0 < /~e-tx = t-l(t.~e -t~) 0),
we have
LAe tAv 2
L2(a)
II -
E/~nAne-tAn~n
-
n=O
12 L2 (Ft)
0(3
-- Z
oo
I~l ~ I~o ~ - ~ I~ o. T h e above argument applies to the case where b o u n d a r y condition (2.3) is replaced by N e u m a n n condition 0u
On
on cgft.
= 0
T h e general case is treated as follows. Let A" realization of
s
D(A) C X = L2(~) ~ X be the L 2
= s162 D)
o :
- E
i,j=l
)a__
~
-~z~a{j(z Oxj + E
o bJ(Z)-~x j + c(x)
j=l
with either (2.3) or
Ou One
+ au = 0
described in w It is associated with the bilinear form A ( , V = HI(f~) or V = H~(f~). We have
D(A) = ( u C H2(f~)
o,,
+a'u=0
(2.11)
) of (1.11) on V x V, where
onOf~
}
if V = Hl(f~), and
D(A) = H 2 ( f ~ ) 2 / H I ( F I ) - {u r H2(f~) [ u = 0
on Of~}
2.1. Evolutionary Problems
51
if V = H~(t2). Consider the parabolic equation
ut + s
D)u = 0
in f~ x (0, oo)
(2.12)
with initial condition (2.2) and boundary condition (2.3) or (2.11). Taking v(t) = e-tan(t) instead of u(t), we can replace A by A + AI for A E R. This allows us to assume the strong coerciveness of the bilinear form A( , ) with a > 0. After the complex extension, we have
a~ A(~, v) > a II~ll~ --
(~
V
V)
(2.13)
(~, v ~ v)
(2.14)
E
besides
I,,A('Lt,,'u)[ ~ C1 I~llv IIv tv
with a constant C1 > 0. The operator A defined in this way is m-sectorial in X. It is of type (0, M) for some 0 E (0, rr/2). Based on inequality (1.16), we can define the bounded operator e -ta by the integration (Dunford integral) for t > 0:
Here, ~ = x/-L--1 and F is the boundary of E0~ = {z 0 < [ a r g
zI
0).
(2.25)
2. Semigroup Theory and F E M
54
In terms of the L 2 orthogonal projection Ph " X --~ Xh, on the other hand, condition (2.24) is equivalent to
~ ( 0 ) = P~0.
(2.26)
Initial value problem (2.25) and (2.26) for Uh is denoted by ( I V P ) h . uh = uh(t) of ( I V P ) h is given by
The solution
~,~(t) = ~-'~,, P,,u0.
(2.27)
For practical computation, one takes the standard basis { ~ h l a E 1-h } of Xh defined by qD2(b) = ~1
(b=a)
[o
(v e Z,, \ {at),
where 1-h denotes the set of vertices subject to ~-h locating in ft. Then, setting Uh(t) ~- E ~h(t)~a, a h aEZh
we determines ~j~(t) through the ordinary differential equation d~
-.
Mh?-/+ K,,~ = 0.
-'*
(2.2S)
a
Here, ~ = (~h I a E Z~,) is an N vector-valued unknown function with N = dim Xh. Mh and /s are N x N matrices defined as follows: Mh = (Mab(h))a,bEZh
with
Mab(h) = (qo~, q~2),
I(h = (I(ab(h))a,bEZ~
with
Kab(h) = ,,4 (qpha,~ ) .
However, representation (2.27) makes it easy to pick up some operator theoretical features of Ah on Xh uniformly in h. We have proven Theorem 1.1 in this direction. Recall that there, crucial facts follow from the consideration on the numerical range. In particular, the operator Ah in Xh is of type (0, M) with 0 E (0, 7r/2) and M _~ 1 indeI)endent of h. The operator e -tAb is well-defined on Xh. Moreover, we have e_tA,, =
1_~27rZ JfF e - t Z ( Z i h -
Ah)-ldz
(2.29)
with the path F in (2.15) and the identity oI)crator Ih on Xh. Consequently, e-tAh Ph = ~ 1
/r e -t~ (ZIh
- Ah)-i Ph dz
follows. A corollary of (2.29) is the estimate
IIAT,~-'~"ll~h,~,, _< n ~ t -~ with Ms independent of h and t > 0, where c~ > 0. The main result of this section is the following theorem. 12,cflccting the smoothing property of parabolic equations, the semidiscrete approximation admits the error estimate of the optimal order for t > 0, even if the initial function u0 is not smooth.
2.2. Semi-discretization
55
T h e o r e m 2.1. Given uo E X = L~(f~), let u and Uh be the solutions (IVP)h, respectively. Then we have
of ( I V P ) and
Ilu(t) - uh(t)llL~(a) + h Ilu(t) - uh(t)iIHl(a) _< Ch2t-llluolls~(a)
(2.30)
f o r t >O. The proof will be done in use of (2.15) and (2.29). We have
eh(t)
=-- u(t) -- Uh(t) ----
27rl f r e-tz [(zI -- A) -1 - (zIh -- Ah) -1 Phi Uo dz,
and therefore, inequality (2.30) is reduced to an estimate on the resolvents. Here we need more detailed studies on the numerical range u(A) of the bilinear form A ( , ). Recall that 0 C (0, re/2) is defined by cos0 = 5/C1 for the positive constants 5 and C1 in (2.13)and (2.14). For 01 E (0, rr/2), the sector is defined by
P~0~ = {~ I 0 _< larg zl _< 01}. L e m m a 2.2. There exists 50 > 0 such that for any v E V and z E C \ Eol , the inequality
Iz Iv ~ + Ilvl ~ _< ~0 Iz vii 2x -
x(v,
v) I
(2.31)
holds. Proof." Take 09 C (0, 0 1 ) and set cos02 = 52/C1. It holds that 0 < 52 < 5. Letting % = 5 - 5 2 > 0, we have ReA(v,v)
- % I1~11~ > > --
cos 0 2 '~ I~11~ ~os o~ 1,4(v,,,)l V
:
C 9 I"
I1,,11~
for v C V. Given ~ c u(A), we have v E V \ {0} satisfying
Call C = ((v). Inequality (2.52) reads;
for # ( v ) = Ilvll~//II~ll~. Therefore we have ~(v)E ~, where
Take z C C \ Eol. Then, an elementary calculation gives dist (z, 7-g) >
Izl sin (01 - -
02) -Jr-~0#(V) sin 02.
(2.32)
56
2. Semigroup Theory and F E M
Consequently we have
Iz Ilvll
- A(v,
v) l :
iz -
(v)l
___ Ilvll~.. di~t (z, ~)
( sin,ol o , osinO
1
2
2
-> ~o (Izl Ilvllx +lvllv)
with 6o = {rain ( s i n ( 0 1 - 0~),%sinO~)}-i > o. This means ( 2 . 3 1 ) a n d the proof is complete. [] An immediate consequence is the following. L e m m a 2.3. Given z E C \ Eol and uo E L2(~), we have
Izl Ilwll~(~) + Izl '/~ I1~1 .,(~) < ~0 Iluol ~(~)
(2.33)
and
IIw H~ 0 and the proof is complete. []
2.4
Full-discretization
In this section we consider approximation methods for (IVP), where the space variable x E f~ is discretized just as in the preceding section, and the time variable t is discretized
2. Semigroup Theory and F E M
62
by a uniform mesh: t = nT with T > 0 a n d n = 0 , 1 , 2 , - - . For the sake of simplicity, we approximate d u / d t in (2.4) by the simplest backward or forward difference quotient. Thus the approximation which we are going to discuss may be called the difference finite element approximation.
Backward
Difference Approximation
First, we deal with the following approximation with backward difference for the time variable:
U~h(t + T) -- u~(t)
+ Ahu~h(t + 7) = O
(2.44)
with u~(0)
=
S;Uo,
(2.45)
where t = n r for n = 0, 1, 2 , . . . The approximate solution u~ is a function from the discrete time t = n r into Vh. It is expressed as
u~(t) : (I,, + ~-Ah)-nP,~,,~o
(2.46)
for t = n r , although for actual computation one discretizes tile time variable in (2.28). Approximate problem ( 2 . 4 4 ) w i t h (2.45)is denoted by (IVP),~,~. Our main concern is tile rate of convergence of ut~ to u. However, we mention the stability of ( I V P ) "r,h u first T h e o r e m 2.6. The fully discrete approximation ( I V P ) T~h u with backward difference in t is absolutely stable. In fact
~ II'~01~(~) Jl,,.,,.(t)llc~(~) r holds for t = nT. This theorem follows immediately because (Ih + TAb) -1 is a contraction on Xh, which is a consequence of
R, (Ah'~,,,,v,,.)= Re A(~,,, v,,.)>_ o. The rate of convergence reflects the smoothing property of parabolic equations. have tile following. T h e o r e m 2 7. Let eh.(t) = "u(t)_. T 9
f o r t = nT.
.
T,h
"
Then we have
We
2.4. Full-discretization
63
Proof." P u t C ( 1 ) ( t ) = u ( t ) -- U h ( t ) = e - t n u o
-- e - t A h P h U o
and ~(2)(t)
=
uh(t) - u~(t) - e-tAhphuo -- (Ij~ + TAh)-nPhUo
--
ts /: Ph u o
for t = nT. We have e~(t) - s(1)(t) + c(2)(t). T h e o r e m 2.1 says t h a t
II~(1)(t)ll.(~)
_< Ch2t-1 ~o11~(~)
for t > 0. In vi@w of IIP~llx~,~ ~ 1, w@ h~v@ only to ~how that
IIK I ~,x~ - I I
+
IXh,Xh~ CTt-1
TAh) -n
(2.47)
for t : nT > O. For this purpose, we write
~ d ((I~ + ~A~)-"~ -"(~-")"~) ~
Z =
n
=
n
sA~(Ih + s A h ) - n - l e - n ( ~ - s ) A h d s 7 ^A3/2
Z
~h
( I h ~- sAh) -(n+
1)
9A h / 2 e - n ( r - s ) A h d 8
and use an inequality, the discrete analogue of (2.16):
IIA~(I~ + ~Ah)-kllx~,~ c~ > 0 and s > 0. This can be proven by means of the Dunford integral in the following way. In fact, letting Fk(z) = (kz)~(1 + z) -k, we have
Fk(sAh)
= - -
(ksAh) ~ (Ih + sAh) -k 2ml f r Fk(sz) (zih
-
where F denotes the p a t h of integration mentioned in w
Ah) -1 dz, For z E F we have
[1 + sz[ ~_ 1 + spcos01 for p = ]z]. If the smallest positive integer greater t h a n c~ is denoted by k0, it holds t h a t
I~ + ~zl ~ >_ 1 + k(k - 1 ) - - - ( k - k0 + 1) (spcosOx)k o k0!
2. Semigroup Theory and F E M
64
by the binomial theorem. We can choose a constant % > 0 satisfying l1 + szl -k 0 a n d p > 0 satisfying
IIA~llx~,x~ (1 + ~) -- ~ < 2cosOl. This is possible because of (2.53). T h e n we i n t r o d u c e a positively oriented c o n t o u r c o m p o s e d of the following two p o r t i o n s (as sets), where R = # / s : F (1)
{re•176 I O p/w = (1 + t~)II&llx.,x~, this implies
II(zI~-A~)-lllx~,~
l+rc < _ R~
__
1
n-IIAhllxh,x~
2. Semigroup Theory and FEM
68 On the other hand we have
ll - szl < I1 - # e ~ < l = (1 + # 2 _ 2 # c o s 0 , ) 1 / 2
= (~2 < 1.
Those inequalities imply IIz(=)ll~,,,x, ' < c
f_o, ( , ~ R ) o < ' (1 + ec) ~ d o = o ( n ~ ) o < ' < c 2 =) 0x
- -
/~/';
- -
(2.57) '
because n~($~ ---+ 0 as n --~ oc by 0 < 62 < 1. Inequalities (2.56) and (2.57)imply (2.55). The proof is complete.
2.5
Inhomogeneous Equation
We consider the semidiscrete approximation applied to the inhomogeneous equation
du dt
--+
Au = f (t)
(0 0)
with
u(0) = u0
Let Vh C V be the finite-dimensional space introduced in ~1.3 and let Xh be the space Vh with the inner product induced from X. We suppose the inverse assumption for the family {rh} of triangulations of 12. The semidiscrete approximation of ( I V P ) is given by
duh + Ahuh = 0 dt
(t > 0)
with
uh(0) = Phuo,
where Ah denotes the self-adjoint operator in Xh associated with AIvh• and Ph : X Xh the orthogonal projection. Let {e -tA }t>0 and {e -tAb }t_>0 be the semigroups generated by A and Ah in X and Xh, respectively. Then,
Eh(t) = e -tA - e-tahph denotes the error operator. We show t h a t L ~162 stability of the approximate solution holds in the following sense. Theorem
2.16.
We have
for Xh E Vh and t > O. For the proof, we introduce the discrete delta function (5 = d~o,h E Vh and the discrete fundamental solution F = r(t) = r~o,~(t) ~ vh:
(~o,~, x~) = r~o,h(t)
=
x~(~o) (x~ e vh) e-~A~o,h.
Given Xh E Vh and x0 C f~, we have (r~o,~(t), x~) = (e-~a~5~o,~, xh) = (hxo,h, e-~A~Xh) = (e-~A~Xh) (ZO)
2. Semigroup Theory and FEM
78 so that
[le--tAhxh LOO(n)O, xoE~
IIr~o,,,(t)ll~, _< C I ] o g h l .
Take the weight h m c t i o n introduced in w
with t~ = 1:
: ~o,h(~) = (1~- ~ol ~ § h ~) 1/2 Similarly to (1.76) we have
I1~-111*L2(a) =
L cr-2dx < C ~ooC rerdr + h2
-
-
c
logh,
and hence
IIr=o,h(t) L, ~
I1~-'11.(~)
9I~=o,hr=o,h(t)llL~(~)
0 is independent ofp > 2. On the other hand, inequality (1.45) is a consequence of the inverse assumption:
[Ix,, [L~(~) < Ch-2/P [IX,,IIL~(n) P u t t i n g p = ]log hi, we obtain (2.79).
[]
We are ready to give the following.
Proof of Theorem 2.16: We show (2.76). First, we have 1d
2
2
= (OtF, o2F) + (VF, V(cr2F)) - 2 (VP crVVcr)
Because
(o~r, >) + (vr, v~) = 0
(~ e v,,),
we have ld
2
2
where
= ( o , r . . ~ r - ~).
n = (vr, v(~r
- ~)),
and I I I = - 2 (VP, a F V a ) .
Let ~b = Ph(a2F) E Vh. Then, I = 0 follows first. By means of tile inverse assumption and L e m m a 2.17, we next get
_< ch-' Ilrllg~(~)h (llrllg~(~) + II~rllg~(~))
Finally,
by I[Vcr[]L~(n) < C. Summing up those inequalities, we obtain d -
-
2
2
2
F[ t,'(a)
81
2. Z L ~ E s t i m a t e
This implies liter(t)
2 IIL~(~)
+
/0 II~Vr(~)ll ~
/0
2
2
and inequality (2.76) is reduced to
~ I r(~) IIL:(~) ds < c laog h
Jj
(2.81)
by Lemma 2.18. Because F satisfies A;IOtF + F = 0
(t > 0),
we have 1 d (AhlF, F ) + IIVll ~
2 dt
L2(a) = O.
Regarding r(O) = 6xo,h, we get 1 (A;1F(t),F(t)) + f0 t
IIr(,)ll~(~)ds
= ~1 (Ahl~zo,h)(X0).
Let
Gxo,~ -- A ~ 16~~,h. Because Ah 1 is positive definite, inequality (2.81) is reduced to Gxo,h(xO) ~llx
W e can estimate the right-hand side of (2.99) from above by
C h - l A I[S- s{~]~ x. We have L 0 is the size parameter,/3 k 0 is a fixed constant, and
D...,(t) = ~-~ {~(t + ~ ) - 2,~(t)+ ,,~(t- ~ ) } For this scheme, Fujii [134] proved tile following. T h e o r e m 2.24. Suppose the inverse assumption and the T / h < t~7 in case 0 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, 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_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
H q~p , ~- U + Yq,~O,
(3.90)
p=l
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 <wb a n d a < l :
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):
The operator E ~ in the right-hand side of (3.84) splits as
n~k E~(tn) =
(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 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. _
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
0 denotes the time mesh size. Based on those facts, we can argue as follows. First, the relation
d--~d (r~(sA)e_n(T_s)A)
=
nr n - l ( s A ) (r'(sA) -4- r ( s A ) ) Ae -n(r-s)A
-- nrn+l(sA)sA2e-n(~-s)A holds, and therefore we have [r~(TA) - e-nrA]A -2 - n
rn+l(sA)se -n(~--s)A ds.
Next, from the semigroup theory, the uniform boundedness of the semigroup represented as
lie-tAll _< C
(0 < t < oc)
(4.1)
is equivalent to the stability of its backward difference approximation:
]rn('rA)ll 0.
r(z) satisfies (iii)o if Ir(z)l < 1 if 0 < Izl < ~ and z E E0 with
If I~(z)l _ 1 for any z E E0, then r(z) is said to be A0-acceptable. Therefore, Aacceptability means A~/2-acceptability in this terminology. Any rational function r(z) of order p (>_ 1) has p r o p e r t y (iii)o for 0 E (0, 7c/2). In fact, for p E [-0,0] we have 0
Op where ~ = x/~-l. Because of r/
r(O) = e-Zlz=o = 1
and
r'(0) =
~-2-e-Z dz z=O
"- --1,
it holds t h a t c)---fi r(Ps
= - 2 R e e~ = - 2 c o s ~
_< - 2 c o s 0 < 0
(4.4)
p=0
and hence the conclusion follows. If r(z) is the Pad~ a p p r o x i m a t i o n of e -z with degrees n and m of the numerator and the d o m i n a t o r , respectively, then it is of order p = n + m. Furthermore, according to n < m, n = m, and n > m, it has properties (i)o, (ii)o, and (iii)o for some 0 E (0, 7r/2), respectively. In fact, in this case we have r(z) = R,,,m(z) = P,.m(z)/Q ......(z) with (n + m - j)!n!
P.,m(~)
=
(~ + .~)!j!(~ _ j)! (-~-Y, j=O
m
Qnm(Z)
(n+m--j)!m!
x-'z_.,(,, + ,~,)b~(,~, - .J)~J j=O
4.1. Rational Approximation of Smigroups
147
The relation
]Rnm(Z)
-
I 0 is a constant and 0>00" 1~ r(z) has property (i)o. 2 ~ r(z) has property (ii)o and 7 IIAI < M~ < +oo. 3 ~ r(z) has property (iii)o and T [JAIl < /l}I~ < ~.
We need several lemmas. constants, respectively.
Henceforth r > 0 and C > 0 stand for small and large
L e m m a 4.2. If a rational function r = r(z) is of order p and Ao-acceptable with p > 1 and 0 E (0, 7r/2), then there exist constants cy > 0 and fl > 0 such that [rn(z) - e -nz ] _< C n zl p+I e -''~'ee(z)
for z C Eo with
Izl
_ ~.
(4.6)
149
4.1. R a t i o n a l A p p r o x i m a t i o n o f Smigroups Proof." We have
rn(z)
--
e -nz ~-- s
r J - l ( z ) ( e - z -- r ( z ) )
e -(n-jT1)z
j=l
and
It(z)
-
e-Z[
0. This implies [r(z)[ _< e -Re(z) + C (Re(z)) p+I for z E E0 and Izl _< a0. T h e function f ( t ) = e - t + C t p+I satisfies f ( 0 ) = 1 and f'(0) < 0, and hence f ( t ) 0 and ~ E (0, cos0). We get
for [zl _ ~)
holds because r = r(z) is rational. Those relation imply rt--1 j=O
4.2. Multi-step Method
153
and hence ~(~) - ~(oo)1 _< ~-~,zlcos0, +
c~-nZ/Izl
(4.8)
if z C E01 and Izl >_ a. Finally, we have
I~(oo)1 ~ e -n~.
(4.9)
Inequalities (4.6), (4.9), and (4.9) give
II~(~-A)II = II 0 is the constant described above as (II)o holds for 0 < z I < ~ and z C Eo.
4. Other Methods in Time Discretization
156
In the second case, min(5, n, 1/lbql ) can be replaced by m i n ( ~ , l / b q [ ) and min((~,n), respectively, if (P, S) satisfies (II)o and bq > O, respectively. Setting
5~(z) = a~ + b~z aq + bqz
(1 _< i _< q),
we can write relation (4.11) as q
5,(TA)un+i = 0
(n = O, 1, 2 , . . . ) .
i=0
First, we study the functions un = u,~(z) of z E C satisfying q
E
5~(z)un+i(z) = 0
(4.13)
i=0
for n = 0, 1, 2 , . . . . We have the following. Lemma
4.9. If ~i(z) (1 0. Lemma
4.10.
rite identity q
e -t"+~A - T~+q(A) = E 7 .... j('rA)Fj('rA) j=o q-1 j (4.16)
j=O k=0
holds for n = O, 1 , . . . Proof." E q u a l i t y (4.15) gives
j=o
j=o z=o n q j=o i=o n+q j=O
4.2. Multi-step Method
159
where J
Bj(z) -- ~
?,-k(z)(~j-k(Z)
(j : O, 1 , . . . , q)
k=O
and q
J~q+j(Z) -- E
"~/rt-J-k(Z)(~q-k(Z)
(j : O, 1, 2 , ' ' " ).
k=0
We have Bj(z) = 0 for j = q , . - . , n + q - 1 from the definition of 7j(z). We also have q
]~n+q(Z) ---
E ~/-k(Z)(~q-k(Z)
--- (~q(Z) = 1,
k=O
so that q--i
e--(n+q)z -- Sn+q(Z) --
~/n-J(Z)PJ(Z) -- E j=o
j
E
~/n-k(Z)(~J-k(Z) [e-Jz -- 3j(Z)]
j=0 k=o
follows. This implies the lemma.
[]
Estimate of
II~xp (-t~+~m) - T~+q(A)[[ is reduced to those of 7j(z) and Fj(z) in this way. For the former we have the following. 4.11. Let (P, S) have property (III)o, and suppose (II)o for 0 < Izl < and z c Eo with t~ > O. Then, each ~' e (O, min(~, 1/Ibql)) admits constants C > 0 and /3 > 0 satisfying Proposition
for Izl < ~' and z E Eo. Proof." As we have seen, the roots Q(z) (1 _< j _< q) of P ( ( ) + inequalities
ICj(~)l _< ~-.,zj
z S ( ( ) = 0 satisfy the
(1~1 _< ~', z e r~0)
for some ~ > O. This implies
IO,n(Z)l < Ce -znlzl
(Izl < ~', z e Eo)
by L e m m a 4.9. Here, the constant C > 0 depends on % , - . . , ~[q-1, which are polynomials o f / ) 0 , " " , (~q-1. The latters are bounded on Izl _< ~' < 1/Ibql, hence C > 0 also is bounded there. Recall t h a t aq 1. The proof is complete. [] :
As for Fj(z) we have the following.
4. Other Methods in Time Discretization
160
P r o p o s i t i o n 4.12. /f (P,S) is of order p(>_ 1) and ec' < 1/Ibq] , then the inequality
IFj(z)l ~ C [zlp+I e-jRez holds for Re z >_ 0 and Iz 0. Recall (4.16). In the second term of the right-hand side, we have
I')'n-k(z)hj-k(z) [e -jk - rJ(z)] l 0 from tile reason described after the proof of Proposition 4.12.
4.2. Multi-step Method
163
Now we give the following.
Proof of theorem 4.8 for the first case:
Because (P,
ICj(z)l _< ~-z~zf
S) has
property
(I)o, the
inequality
( z l _ ~', = e zo)
holds for arbitrarily large ec' > 0 with s o m e / 3 > 0. T h e conclusion of Proposition 4.11 still holds for any ec' >> 1. M a k i n g / 3 > 0 smaller, we have
~j(z)l _< ~-~ < ,
( =1_ ~', = ~ x0),
I~'~(~)1 _< c~ -n~
(z >~', z~Eo)
(4.18)
so t h a t
follows as in the proof of Proposition 4.11. To estimate the first t e r m of the right-hand side of (4.16), we take M > r IIAII and represent n
1
+ ~r ) s %_j(rz)Fj(rz)(zI-A)
E %_j(rA)Fy(rA) = j=0
2 7"i-~
i nt_l-,2
3
-1
dz
j=0
= I+II. Similarly to the case (b), we have
I-Zll _< o~-p with the constant C > 0 independent of M. If j _> 1, we get
Vn_j(~z)Fj(~z) (~Z- A) -1 & _ R, z e r,o)
4. Other Methods in Time Discretization
164
for R > 0. Actually, from the proof of Lemma 4.9 this is reduced to e-~n
I~j(z)"- ~j(oo)nl ~ c ~
where ~j(z) (1 _< j _< q) denotes the root of P(r ~(z) o - ~(oo).l
:
(1=1 _> R, z ~ ~o),
Izl
+ zS(r
,~3(z) - ~ ( o o ) ,
(4.19)
= 0. Inequality (4.18) implies
n-1 . }2 ~;-~-~(z)~y(oo) k:0
o and {exp ( - t B ) } t > o satisfy
)in lim [r ( t ) ( o r n--,+c~ nt m
=exp(-t(A+B))
where r = exp ( - t A ) and r = exp ( - t B ) . H.F. Trotter extended this property to (Co) semigroups on Banach spaces. It has been called Trotter's product formula. Error estimates in the operator norm were known for bounded operators, but only strong convergence has been discussed for the other cases. However, D.L. Rogava has succeeded in giving them for analytic semigroups. We shall describe the simplest case. T h e o r e m 4.13. Let A1 and A2 be positive self-adjoint operators on a Hilbert space H satisfying IIAll/2e-tA2All/21 < C
(4.21)
for t e [0, J], and suppose that A = A1 + A2 is self-adjoint with D(A) = D(A1) N D(A2) and furthermore, D(Aal/2) C D(A~/2) N D ( A 3/2) holds. Then, T ( t ) = e-tA2e -tA1 satisfies the estimate
sup
tC[O,nJ]
IIe -tA -- T ( t ) n l l
with a constant C > 0 independent n = 1, 2, 3 , - - . .
< C---~ lx/'n
(422)
4. Other M e t h o d s in T i m e Discretization
166 This t h e o r e m bounded domain "10a = 0, and A2 We take a few
is applicable, for example, if A1 is the differential operator - A on a ft C R n with s m o o t h b o u n d a r y 0f~ provided with the b o u n d a r y condition = V ( z ) >_ 0 with V ( z ) sufficiently smooth. preliminaries. Let
A =
fO ~
AdE(A)
be the spectral decomposition of a positive self-adjoint A operator in H. t >_ 0 and c~ E [0, 2], we have
First, given
OG
[(I + t A ) -1 - e -tA] A -~ =
[(1 + tA) -1 - e -t~] (At) -~ d E ( A ) , t ~.
Because sups>0 I[(1 + s) -1 - e -~] s-~l < +oc, this implies
II [(I +
t A ) -1 - e-tA]
A- II
ct~.
(4.23)
(s > 0).
(4.24)
We also have tile elementary inequality (1 - e - S ) -'/2 0
for t > 1.
Inequality (4.24) implies (1 - e - ~ ) -1 d lE(A)'u 2
=
0}. In this case OWa/OXl = 0 and hence 0w~ 0Wb V w ~ . Vwb = Ox2 " Ox2 follows. Let {Pj [ 0 < _ i < _ 2 } be the vertices o f T with P0 = a. Because T i s a r i g h t or acute triangle, the vector e = ( 0 , - 1 ) is expressed as 2
e = ~_s aj eoj j=l
with some aj >_ 0, where e0j = P 0 ~ . j = 1, 2. It holds t h a t 0W a
Ox2
Here e 0 j . Vw~ is a negative c o n s t a n t on T for
2
= e . VWa = ~ a j e o j j=l
9VWa 0 (j = 1, 2) if b -r a, and
Owb > 0 O3g 2
--
holds similarly on T. Therefore, inequality (5.13) follows and the proof is complete. We are now ready to give the following.
Proof of Theorem 5.1: Note that (5.8) is reduced to A > 0, fh E Xh
===> m_ax(Ih + AK,71Ah) -1 fh _< max{0, m_ax fh}. fl
(5.14)
f2
Here, the well-definedness of (Ih + AKs
-1 is also included. In fact, (5.14)implies
fh _ 0.
or equivalently,
Then, tile fundamental relation
e-tKs
Ah
=
lim (Ih + tn- 1Ks lAb) -n 71,'----+00
assures (5.8). L e t [ . ]+ = m a x { . , 0 } .
(5.1,5)
Property (5.14)follows if max_ Uh _< max_ rrh [fh]+
(.5.16)
is proven for Uh C= Xh and fh = (Ih + AI(s In fact, then fl + = 0 implies u h+ = 0 and hence fh = 0 gives 'uh = 0 is obtained. This means the well-definedness of (Ih + AI(,71Ah)-I " X,, ---+Xh and also (5.14). For (5.16) to prove we may suppose that the maximum of the left-hand side is attained at some a E :2h with a non-negative value, because the right-hand side is non-negative. Suppose those situations. It holds that
(a~, ~h) + A (v,,,h, Vx~) = (L, ~,~) for Xh E Xh. Let Xh = IDa"
(a~, tea)+ A (v,,,,,,, W*,a) = (/,,, r&) We have (gZh, g&) = uh(a)Isupp Wal
and
(-fs ~ )
= fh(a) supp ~ l .
(5.17)
5.1. Lumping of Mass
177
Furthermore, the equality (VUh' VWa)
---
E Uh(b) (VWb, VWa) bEZh
=
~
(~(b) - ~ ( a ) ) ( W ~ , W a ) + ~.(a) ~
bEZh
--
(W~, W a )
bEZh
~
(~th(b) -- ,/th(a))(VWb, VWa) -- "/th(a) ~
bEZh\{a}
(V~Ub, Y e a )
bEI3h
holds by (5.9). Because a E/-h attains a non-negative maximum of uh we have (Vuh, VWa) ~ 0 by (5.12). Those relations imply
m~x_ ~ = ~(~) 0
(5.31)
(BhUh, Xh) = supp Wa] (b(a). Vuh)r(a) --> 0.
(5.32)
and
Because the right-hand side is non-negative, the left-hand sides of (5.31) and (5.32) are equal to 0. If the first assumption holds, then a~ attains maxg'uh for i = 1, 2 , . . . , m + 1. This implies m a x ~ u h _< 0 by 'uh = rrhg _< 0 on oqf~. If the second assumption holds, we have (VWa, Vwb) < 0 for a : / b . Then, the inequality (5.31) implies that 'uh is a constant on f~. Then m a x g u h _< 0 follows similarly. To prove the latter part, we take A > 0 and u~, E Wh satisfying u h - rrh9 E Vh and
(~h, ~,~) + a (v,,,,h, vxh) + (~h,,,,, ~,~) = (~-Tf, ~,~)
(5.aa)
for )C~ E Vh. Without those assumptions, we can show m_ax a uh _< Inax { m_ax a rrhf, I ~ X rrh } .
(5.34)
In fact, if a E 5~ attains max~u,, we have (B,,u,,, ~,,) >_ O. Then, relation (5.33) implies (5.34). The proof is complete. []
5.3
Mixed Finite Elements
The m e t h o d in consideration is concerned with the variational problem with constraints. To describe the idea, we take the Dirichlct problem for the Poisson equation Ap = f
in [2
with
p = 0
on cgf~,
(5.35)
where f~ C R 2 is a polygon. Putting 'u = Vp, we llav(; u-
Vp = 0
and
V . ',, = .f.
(5.36)
183
5.3. Mixed Finite Elements
Let V = L2(f~) 2, W = H~(~), and B ' = V : W ~ V. The dual operator B = ( B ' ) ' : V "~ V' --, W _~ W' is defined through the representation theorem of Riesz. Similarly, f 6 L2(ft) determines 9 e W', regarded as an element of W, by (g, q)w = (f, q)L2(fl) for q 6 W. Under those preparations, (5.35) has an abstract form, to find (u,p) 6 V x W satisfying
u + B'p = 0
B u = 9.
and
(5.37)
As' we shall see, this problem is realized as an Euler equation on u for a variational problem with a constraint, where p acts as a Lagrangian multiplier. Then, it is transformed into another variational problem without constraints, where the solution (u, p) is to be a saddle point. Method of mixed (hybrid) finite elements arises naturally as a discretization of those structures. Although it had been applied to many problems in engineering related to solid mechanics, hydrodynamics, electro-magnetic theory, and so forth, the discovery of such a fine structure made it much more reliable practically. This section describe the fundamental ideas.
Abstract
Theory
We take the abstract theory first. Let V and W be real Hilbert spaces, and ,4 and B be bounded bilinear forms on V x V and V x W, respectively. We have
1"t4(u, ~)1 ~ C1 for u , v E V, where C1 > 0 is a constant. variational problem with constraints,
II~lIv IIvll~
Given g c W and F E V', consider the
(5.39)
inf u ,7, where U = {v E V l B ( v , q ) =
(5.38)
(9, q)w for any q E W} and
y(~)
= ~
A(v, ~) - F(~)
The bounded linear operator B : V ~ W is defined by
(B~, q)~ = z(~, q)
(q ~ w ) .
We have U = {v E V I B y = g} and hence U # 0 if and only if g 6 Ran B, which means the existence of Uo E V satisfying Buo = g. Then it holds that U = {u0} + Ker B. It is a closed atiine space in V and the s t a n d a r d argument described in w the condition
A(v, ~ ) ~ ~ I vll~
(v
e
Ker B)
applies. If
(5.40)
5. Other M e t h o d s in Space Discretization
184
holds with a constant a > 0, variational problem (5.39) is uniquely solvable. The solution u C V is characterized by u-
u0 E U
and
.A(u, v) = F(v)
(v C Ker B ) .
(5.41)
Let c~ : V' + V be the canonical isomorphism defined through Riesz' theorem, and A : V + V the bounded linear operator associated with A:
A ( ~ , ~) = (Au, ~ ) ~
(~, ~ 9 v ) .
Tile second equality of (5.41) is expressed as
(oF-
Au, v)v = 0
(v 9 Ker B ) ,
or equivalently, a F - A u E (Ker B) • in V. Here, the dual operator of B is realized as B' : W + V through Riesz' theorem: V' " V, W' _~ W, and then (Ker B) • = Ran B' follows. Under the assumption that Ran B' : closed
C V,
(5.42)
u solves (5.41) if and only if it admits p E W satisfying
A u + B'p = o F
B u = g.
and
(5.43)
Note that p acts as a Lagrangian multiplier here, and (5.37) obeys a form of (5.43). If one prefers to tile weak formulation, (5.43) may be replaced by
.A(u, v) + B(v, p) = F(v)
B(u, q) = (g, q)w
and
(5.44)
for a n y v C V a n d q C W . Problem (5.43) is reduced to another variational problem without constraints. In fact, (u, p) solves (5.43) if and only if it is a stationary point of 1
,]('u, q) = -~A(v, v) - F ( v ) + (q, B v - g)w
(5.45)
on V • W. Actually (u, p) is a saddle point,. To see this, take q E W and v C V. If (u, p) solves (5.43), then we have 2(,,,, p) - 2(,,,, q) = (~, - q, B,,, - 0 ) ~
= 0
and
! A ( . . , .~,)- ~A(.,,,, . . ) 2
F(.v-
u)+
(~, B ( v -
~))~ 1
= A(.,~, v - u) - r(,,, - ..,) + (p, B ( v - ..,))~ + ~ A ( v - .,,,, ,,, - u) 1 A ( , v - ,u, v - u) a F , v - "u)v + -~
=
(Au + B'p-
-
_1 2 A (v - u, v _ u) > O.
185
5.3. Mixed Finite Elements
In particular, the relation 2 ( u , q) _< 2 ( u , p) _< ST(v, p)
(v e V, q E W)
(5.46)
follows as is expected. Relation (5.42) is equivalent to the existence of k > 0 satisfying
IlB'qllv =
sup ~o~
> kllq ~li~
-
for q C (Ker B') -L. Because (Ker B') • = Ran B, this means sup
B(v, q) > k Ilq w
(q E Ran B).
(5.47)
v~.\{0~ II~ll. -
Here, the closed range theorem says that Ran B' C V is closed if and only if Ran B C W is so. Because (Ker B) -c = Ran B', similarly it is equivalent to the existence of k > 0 satisfying
sup
qeW\{0}
15(v, q) > k II~llv q Iw --
(v e R a n / 3 ' ) .
(5.48)
We have the following. T h e o r e m 5.6. Let V and W be Hilbert spaces over R, and ~4 and 13 be bounded bilinear forms on V x V and V x W , respectively. Suppose (5.38), (5.~0), and (5.47) (or equivalently (5.~8)). Let A : V ---+ V and B : V ---+ W be the bounded linear operators associated with A and 13, respectively, and take g E Ran B. Then, the solution (u, p) r V x W of (5.~3) (or ( 5 . ~ ) ) exists, and is characterized as a saddle point of 2 defined by (5.~3). Here, u C V is unique, while p E W is unique up to an additive element in K e r B ' . Actually, p is taken uniquely in Ran B, and then we have Ilullv + IIPlIw < C (IIFIIv, + Ilgllf),
(5.49)
where C = C (5, C1, k) > 0 is a constant. Proof: From the above arguments follow the equivalence between (5.46) and (5.43), the existence of the solution (u, p), and the uniqueness of u. The orthogonal decomposition u = Ul + u2 with Ul C Ker B and u2 C Ran B' gives Bu2 = g, and assumption (5.48) assures
I1~11~ _< k'-I Ilgll~. Furthermore, A u l = ~ F -
Au2 - B'p and hence
A(ul, v) = f(?3) -- A(u2, v) follows for v E Ker B. Putting v = ul, we obtain
II~lll. _< ~_1( FII., + 61 lu~ll.) _< ~-1 (llrl ., + c~k-' I gll~).
186
5. O t h e r M e t h o d s in Space Discretization
Similarly, the orthogonal decomposition p = Pl + P2 with Pl E Ker B ~ and p2 E Ran B gives B~p2 = o F - A u and Ilp211w 0 independent of h. Proof: Equalities (5.43) and (5.57)-(5.58)imply A ( u - Uh, Wh) + B(Wh, p - Ph) = 0
and
13(u - Uh, rh) = 0
for any wh E V~ and rh E Wh. Taking vt~ E Vh and qh C Wh arbitrarily, we have A(v~ - uh, wh) + S(w,~, q~ - p~) = A(vh -- u, wh) + S(w~, q~ -- p) and B(~
-
~,. #o = C(7)Ilbll 2L ~ ( n ) " Here, C(7) > 0 is a constant determined by the parameter 7 > 0 arising in connection with the regularity of {%}. See also [375]. 5.3. The inverse operator of the canonical injection a : V' --, V is sometimes referred to as the duality map. Note that the bounded linear operator A : V --, V is different from that of the previous sections defined through the Gel'fand triple V C H C V'. Variational problems of saddle point type arise in many areas. See Aubin and Ekeland [15], Rabinowitz [320], and Suzuki [369]. In connection with H(div, f~), the function space H(rot, f~) is also introduced when f~ C IR3. It is composed of the vector field 'v E L2(t2, R 3) satisfying V x v E L2(t2, Ra). See Girault and Raviart [160] for those spaces and applications. The existence of v satisfying (5.54) is also proven there. See also L e m m m a 7.32 in w Instead of (5.52) with (5.53), the Stokes system may be formulated as to find (u, p) E H(~(f~) n x L2(t2) satisfying 2~
(e~j('u), e,j(v))L~(n ) - (p, V 9v)L:(n ) = (.f, v),:(n),~
i,j
for ~ c H ] ( ~ ) n ~nd ( V . ~,, q)~(~) = 0 for q c L : ( ~ ) , w h ~
1 (O'u ~ Ou~) r
= ~ \o.~, + ~
Colnmelatary to Chapter 5
205
This way is more natural from the physical point of view. Then, Korn's inequality takes place for Poincar~'s one to confirm the assumptions of Theorem 5.6. The continuous Stokes system and its finite element discretization were studied by many people, including Cattabriga [70], Kellogg and Osborn [217], and Crouzeix and Raviart [101], Bercovier and Pironneau [30], Glowinski and Pironneau [163], LeTallec [237], Stenberg [357], respectively. Mixed finite elements for magnetostatic and electrostatic problems were studied by Kikuchi [219], [220], [221], [222], [223]. Concerning the scalar equation, the Stokes system, and the equation of linear elasticity, see Brezzi, Douglas, Jr., and Marini [67], Raviart and Thomas [325], Crouzeix and Falk [99], Crouzeix and Raviart [101], Falk [122], and the references therein. Inequality (5.56) is referred to as Babu~ka-Brezzi-Kikuchi's inf-sup condition. It was introduced by Babu~ka [18], Kikuchi [218], and Brezzi [66] independently. Concerning actual examples of Vh and Wh satisfying the assumptions of Theorem 5.7 in use of finite elements, see Giraut and Raviart [160] and Brezzi and Fortin [68]. Under this condition, the discretized nonstationary Stokes system is treated similarly by the semigroup theory. See Okamoto [307] for details. The Navier-Stokes system is the fundamental equation in fluid mechanics. Operator theoretical approach for nonstationary problems was initiated by Kato and Fujita [207] and Fujita and Kato [143]. For other references, see the commentary to w Its finite element discretization was studied by Okamoto [308] and Heywood and Rannacher [175], independently. Error estimates given by the former are a priori; they hold only under reasonable assumptions on the initial value. The latter's are, on the other hand, a posteriori so that are valid under some additional estimates of the solution. Up to now, fundamental theory for this system is unsatisfactory, especially for the case of three space dimensions. A posteriori approaches are intended to compensate such a situation by numerical computations. See also Bernardi and Raugel [34], Heywood and Rannacher [176], [177], [178], and Rannacher [322]. 5.4. For layer potentials, see Courant and Hilbert [94], Kellog [216], and Garabedian [155]. Theorem 5.8 was proven by LeRoux [233] for n = 2 and Nedelec and Planchard [288] for n = 3. We have followed the method of Okamoto [309] for the proof, where q E X is taken in L2(cgf~). In the actual computation, the method of collocation is more realistic than that of Galerkin's. Iso [191] and Hayakawa and Iso [168] made error analysis for this method applied to the Neumann boundary value problem. Related works were done by Arnold and Wendland [13], [14]. Several monographs and surveys were published concerning the boundary element method. See Sloan [350] and the references therein. For the theory of distributions, particularly equalities (5.72) and (5.76), see Yosida [410] and Schwartz [348]. 5.5 CSM is also called the fundamental solutions method. Since the pioneering work by Bogomolny [41], the efficiency of the scheme has been clarified rigorously. Theorems 5.10 and 5.13 were obtained by Katsurada and Okamoto [213]. Theorem 5.14 was proven by Katsurada [210]. For other charge and collocation points, it can happen that the
206
5. O t h e r M e t h o d s in Space Discretization
coefficient matrix G becomes singular. Some examples are given in [210]. Problem (5.63) is invariant under the following transformation: (1) x H c~x, yj H c~yj, where c~ is a constant (2) f (x) ~ f (x) + c and v ( x ) ~ v ( x ) + c, where c is a constant. Scheme (5.77) has lack of such properties. K. Murota proposed N
j=l
where Q o , ' " ,
Q N are determined by VN(Xj) = f ( x j )
(j=
1,...,N)
and N
k=l
This scheme is provided with such properties. In this case, tile conditions R g - oN ~ 1 and R ~ 1 are not necessary for the wellposedness and the error estimate to hold. Katsurada [211], [212] studied the case that c%2 is a Jordan curve in use of the conformal transformation. An application to the free boundary problem was made by Shoji [351]. Furthermore CSM is a powerful method to obtain numerical conformal mappings. See, for this topic, Amano [6], [7], and a m a n o et. al [8].
Chapter Nonlinear
6 Problems
Many problems in applied sciences and engineering are formulated as nonlinear partial differential equations. Fortunately, in accordance with the progress of analytical theories, reliability on numerical computations has been advanced extensively. In this area we have several monographs such as Girault and Raviart [159], [160], Kf{~ek and Neittaanmgki [227], and Zen{gek [417]. This chapter selects the following topics: (1) iterative method for solving unstable solutions for elliptic boundary value problems, (2) qualitative features of finite difference solutions for semilinear parabolic equations, (3) finite element scheme for degenerate parabolic equations based on tile L 1 structure.
6.1
Semilinear Elliptic Equations
A typical example of the nonlinear elliptic boundary value problem is
-Au-
lul p-1 u
in ~
(6.1)
with u = 0
on 0f~
(6.2)
for p E (1, ec). In this problem, u = 0 is the trivial solution and non-triviM solutions are linearized unstable so that the simple iteration scheme U k + 1 ~- ( - - Z ~ ) - 1
I~1 ,-~ ~
(6.3)
hardly converges to them. Furthermore, the existence of non-trivial solutions depends sensitively to the domain: dimension, topology, and geometry. To fix the idea, let ~ C R 2 be a convex polygon. Then, it is known that problem (6.1) with (6.2) has infinitely many solutions. The functional
J(v)-
3f~(1 ~IWl ~-
1 p+l
I~
,p+) '
d~
takes place of the energy, and u is the solution if and only if it is a critical point of J on
v = H0~(~). A least energy solution denotes the one which attains the minimum of J on the set of non-trivial solutions. It is positive in g2 and attains d = inf J > 0, H 207
(6.4)
6. Nonlinear Problems
208 where
c~(a) Note that J(v) =
(1 1) ~-v-4-r
--
Ilvl
Lp+l(f~)
IlWllL2(n) for v 9 H .
>0
"
We call (6.4) Nehari's variational
formulation and N" the Nehari manifold. By Sobolev's imbedding theorem we have inf I WilL2 > O, yEAr (fl)
(6.5)
and this implies d > 0. It is known that d is equal to the mountain pass critical value so that d is attained by a solution with the Morse index 1 if it is not degenerate. Subject to the underlying variational structure, the iterative sequence {uk}k=0,1,2,... in consideration is constructed on iV" in use of the scalar multiplication operator Q" V\{0} ---. Af. Actually, it is not hard to see that any v 9 V \ {0} takes unique t = t(v) > 0 satisfying Q(v) - t(v)v 9 N'; t h a t is,
t,_l = II Ilvl ,+1 Lp+l(gt)
Then the sequence is defined as Uk+l = Tuk for T v = Q ( - A ) -1 Ivlp-lv and u0 9 V \ {0}. More precisely, Uk+l
-~
tkwk 9 Af
with
tk > 0
and
wk = ( - A ) -~ ]'uk p - 1
Uk.
(6.6)
Here, the positivity is preserved so that if Uo _> 0, then 'uk > 0 for k = 1 , 2 , . . . This sequence is provided with several fine properties. First observation is the following. L e m m a 6.1.
We have t(v) 0
(6.11)
IIw p+l Lp+, - 2 ( V w ~ , v u ~ ) +
IlW~ll =L2(a) + o(1)
211
6.2. Semilinear Parabolic E q u a t i o n s
holds. Here, we have
(Vwk, Vuk)
---
I uk p+~1
---
V?~k
~ 2(a) L
so that l i m [IX7 ( U k + 1 - k---,oo
Uk)llL2(f~) -- 0
follows. In use of the bootstrap argument for (6.10), this implies lim Iluk+l - uk w -- O.
k----~oG
Now, suppose that a~(u0) C W is not connected. Because it is compact, then we have non-empty disjoint compact sets A and B satisfying cJ(u0) = A U B. If dist (A, B) = 35 > 0, we have N such that I]Uk+l- ukll~y < 5 for k > N. From the definition, there exists kj ~ oc and k~ > kj satisfying dist (uk~,A) < 5 and dist (uk~, B) < 5. Therefore, there exists some kj" in k i < kj" 0 and p E [1, oc]. Here, it should be noted that all (1 _< p _< ec) are equivalent for h fixed. Actually we have
[lun[IL~(a)
1
[lu"llL,(a) < Ilu"llLOO(a) < h - ? ]lU n I]L,(a)"
(6.14)
Then, the scheme in consideration is the finite difference m e t h o d (FDM), implicit in linear parts, n
. n+l
=
+ (u;) 2
he
Tn
(6.15)
for j = 1, 2 , . . . , N - 1 and n = 0, 1 , 2 , . - . with
u~=u}=0
(n>0)
(6.16)
and
uj0 =Uo(Xj)
(l_<j < N-l).
(6.17)
Scheme (6.15) with (6.16) and (6.17) approximates the solution and the blowup time appropriately. We describe the qualitative features of the approximate solution in details. For this purpose we suppose that Uo(X) is symmetric with respect to x = 1/2 and N = 2m so that u 0 ( 1 - x ) = uo(x) and Xm = 1/2. We also suppose that Uo(X) is not constant and monotone increasing in [0, 1/2]. Then it follows that
u~_j = u2
(j=l,2,...,m;
n=0,1,2,...)
and n
(j=
1,2,.-.,m-i;
n=
1,2,..-).
In particular, we have and
n ?~m-
l --- I n a x
n uj.
j~=m
We suppose that the solution of system (6.15) with (6.16) and (6.I7) blows up so that oo
lira
n----* (x)
II'u'~IIL~(~)-- -t-c~
and
~
w,, < +o -
-
n "~nun+l -It- Urn-1
n+l
and
l+2)~n
um
> -
n Urn l+2A,~
,
respectively, and hence
~u~/I ~IIL~(~) + (1 + 2.Xn)u~_ 1 (1 + 2:~.) ~
~:+1 >_ ~ . ~ + (1 + 2 ~ . ) ~ _ ~ (1 + 2)~n)2 follows. Because
b~ ~
, n ~m u~[ LP(fl)
>_1,
we have n
~21 >
,~ + u r n _ 1
(1 + 2s
2
6. Nonlinear Problems
214 so t h a t relation (6.18) follows as II
lira inf U mn _ n~oo
1
= lim inf u~ +11 _n 1. _ > ~ lira inf A + u,~_l ~_ A + lira inf um n-.o~ n-~oo (1 + 2An) 2 n-~oo
In use of (6.21) and (6.22) we have
=
An (U~n+'l + U~n+') + [1 + %U~n_,] U~n_, 1 +2An (1 + 4An) Anun+21 -Jr An [1 + "rnu~] u,~ (1 + 2A,~)[1 +
(1 -4- 2An) 2
+
TnU,nn_l] tt,nn_l
(1 + 2A,~)2
This implies l t n t 11
1 for n large, and then T,, = r/I1~11~(~) follows. Letting an --
, n /tm_ 1
n
UTn
'
we have by (6.23) that u~+ll an+ 1
--
(0.24)
7tn+ 1
(1 + 2~,,)
2A,, + [1 + T,,'u~] l t mn ,/ l t mn+l - 1
A. [I + ~-,,,~] u~ + (1 + 2;~,,)[I + ~',,~,m_,] '~ Urn,
-
Irt
i,~
n
i
n -t- 2A,, [1 -t- T ,,.-~m~ln 1 ] "ltm.(1 + A,,)[1 § T,u~] u m " 1
A (1 + rb,,) + (1 4- 2A,,) [1 + rb,,a,,.] a,,
(1
+
~..) (1
+
~b..)
+
2~.. [1
+
(~.25)
~..~,...] a..
In other words, we have
*"('+'~") + (1 + 2A,,)[1 + rb,~a,~] a,,
an+ 1 ~
(6.26)
a,,. - (1 + A,,.) (1 + rbn) + 2A,, [1 + rb,~a,r a,. for n large. To get a reverse inequality we reduce (6.21) and (6.22) as -)',n'~'n+ 1 . . m . -{-
1 -Jr-TnH,,nn_ 1 ILm_ 1
1 + 2A,,. 2 -.%u.,,._, -2, n+l
, n 'u;',,. § (1 + + A,,~ [1 + r,,um]
2,~,u)
[1 §
'" 1 Tn ll'Tn.- 13"II'm-
(1 + 2A,,.) 2
> -
A.. [1 + ~..,,........]. u .... " +(1+
2A,.) [ l + % u ' "..... , ] ' 'bl'm- 1
1 4- 4A,, + 4A~,
....
215
6.2. Semilinear Parabolic Equations Similarly, we obtain an+l
n Um-1 un+l
=
(1 + 2/~n) un+21
~+~1 2,~ a m_
+
[1 + r
n um
a~ [1 + ~n~ n] ~ + (1 + 2a~) ~ [1 + ~ n ~ _ , ] ~ _ ,
>
-
(1 + 2A~)[1 + ~ , ~ ] ~m + 2A~ [1 + ~ ; ~ - 1 ] ~ - 1 an (1 + ~b~) + (1 + 2a~)[1 + ~b~a~] a~ (1 + 2An)(1 + 7bn) + 2An [1 + Tbnan] an
for n large. This means t h a t an + l ~
an
ar~
- (1 + 2An)(1 + Tbn) + 2An [1 + Tbnan] an
(6.27)
We can deduce t h a t {an} is m o n o t o n e decreasing for n large by (6.26). Actually, t h e n 2 {)~n (1 + 7bn)+ (1 + 2/~,~)[1 + 7-bnan]an}-{(1 + )~n)(1 + 7bn) an + 2~n[1 -t- 7bnan]an}
= )~n (1 -- an)[1 + Tbb -~- 2an (1 + Tbnan)] -~- Tanbn (an -- 1) is negative. Because
0 < an < 1
and
1 1 _< b~ _< b = h 5,
(6.28)
this t e r m is d o m i n a t e d from above by
7n (1 -- an) [--Un_l -~- 3h -2 (1 nt- Tb)]. We have proven (6.18), so t h a t there exists k satisfying k > Urn-1 -- 3h-2(1 + Tb) > 2h -2. Relation (6.21) implies
un4-1m'-i> -- --
1+
n
TnUm-1 n Urn--I"
l+2)~n
Therefore, we conclude Unmt21 ~ U n _ l ~ 3 h - 2 ( 1 + Tb)
for n >_ k by an induction. T h u s {an}n>k is m o n o t o n e decreasing. Let a = l i m n _ ~ an C
[0,1). By (6.28), there is a subsequence {bin} C {bn} converging to some constant c _> 1. Then, (6.25) gives t h a t
a < 1 +7-ca I+TC
.a.
6. Nonlinear Problems
216 This implies a = 0 and hence lim,~__.~ b,~ = b. Furthermore, lim An = lim "rbnh -2 ~--~ a~ ~--~o~
( [lt~m, _ l )
-1
~-- 0
by (6.18). In use of (6.26) and (6.27) we obtain lim bn -- lim a~+l -1 E (0, 1). n---<x~ an 1 + ~-b
(6.29)
n---*cx~
In particular we have oo
oo
Ea,~
< +ec
E(l+Can) 0 admits an order preserving contraction (I + AA) -1 in X. Abstract theory of M.G. Crandall and T. Liggett now guarantees the convergence
S(t) = s-lim m--,oo
(
I +
)m
t_ A T~
(6.33)
and thus the nonlinear semigroup {S(t)}t>_o is defined on D(A) = X . Then, u(t) = S(t)u0 is regarded as a solution of (6.30). Semigroup {S(t)}t>o has the properties of order preserving and L 1 contraction, described as (6.34)
218
6. Nonlinear Problems
for Uo, fto C X and t >__ 0, where v+ = max{v,0}. (I + AA)(X) = X, and
This is a consequence of (6.33),
where v,~ 9 D ( A ) and A > 0. Time discretization of (6.30) was studied by A.E. Berger, H. Brezis, and J.C.W. Rogers. Based on the nonlinear Chernoff formula, they developed the L 1 theory. Here, we take into account of the space discretization, and consider the finite element analogue for (6.32). Decomposing ~ into simplexes with the size parameter h > 0, we write Th for their totality. Furthermore, Xa denotes the set of continuous functions on f/, linear on each T r rh, provided with the topology induced by LI(~). From the L 2 theoretical point of view, it is natural to take d d~(Uh, ~h) + (Vf(u,,), V~a) = 0
with
as a semidiscrete approximation, where r
(uh(0), f#,) = (Uo, Ch)
(6.35)
r Xh. The solution
uh 9 C I ( [ O , T ) , X , ) may be regarded as a finite element approximation of u. In terms of the operator theory developed in the previous chapters, scheme (6.35) is represented in the following way. First, the finite element approximation Lh" Xh ~ Xh of L = - A and the Ritz operator Rh" V ---+ Xh for V = HI(U) are defined as follows, where A(u, v) = (Vu, Vv)" (i) L,,vh = fh r (ii) Rhv = v,, r
A(V,,, ~ h ) = (fh, fa,,)
for fa,, e Xh ;
A(v, fh) = A(v,,, gab)
for fa,, 9 X h .
Then, (6.35) is written as
duh
d----t-+ LhR,,.f('u,,)= 0
with
"u,,(0)= P,,u0,
(6.36)
where Pt, " L2(f ~) ---+ Xh denotes the orthogonal projection. If f 9 IR --+ IR is locally Lipschitz continuous, scheme (6.36) is well-defined, because then vh E X,, implies f(v,,) E V. Scheme (6.35) is conforming only in this case; otherwise it is not so. We propose another scheme, provided with the properties of L 1 contraction and order preserving. Such a viewpoint is not always regarded so significantly in the linear case f ( u ) = u, but becomes important in the degenerate case as we shall see. It is actually realized by replacing the Ritz operator by that of interpolation, and adopting the technique of lumping of mass for the linear part. Let Fh be the set of vcrtices subject to rh. For a E Yh, the function wa C Xh is defined by
W a
1
ata
0
at b E F h \ {a}.
~---
6.3. Degenerate Parabolic Equations
219
Then, {Wa I a ~ v . } forms a basis of Xh and the interpolation operator 7rh: W = C(f~) Xh is defined by
7ChV= E v(a)Wa. a6"Ph Remember that ~ C R 2 is a fiat torus so that is compact by itself. We adopt the m e t h o d of lumping of mass described in w Actually, each a 9 )2h takes the barycentric domain Da. From the periodic extension, it is identified with a subset of ~. Let
(x 9 Da) (x 9
1
Wa(X)--
0
,
and denote by X h the vector space generated by {~a [ a C ~2h}. The lumping operator Mh " Xh --~ X h is defined through We H ~a. The adjoint operator M~ 9Xj~ ~ Xh is associated with the L 2 inner product, and we set /(h = M ~ M h " Xh ~ Xh. Let u0 E W. The scheme studied in this section is described as d dtKhUh + LhZChf(Uh) = 0
with
Uh(0) = 7rhUo
with
(Uh(0), Wa) = (TChUo,Wa)
(6.37)
in Xh. In the weak form, it is written as d d--t (Uh, Wa) + (VWhf (Uh) , Vwa) = 0
for a E 12h, where Uh = MhUh. Because 7chf(Uh) E Xh, it is conforming for any continuous non-decreasing function f satisfying f(0) = 0. Furthermore, the linear operator 7rh " W = C(f~) ~ Xh is order preserving and so is the mass lumping Mh 9Xh ~ Xh. This is desirable for our purpose. We suppose that any T E Th is a right or acute triangle. This implies m a x i m u m principle and L 1 contraction for the discretized linear part. In fact, any result stated in w is concerned with the homogeneous Dirichlet boundary condition, but similar arguments assure them for the periodic b o u n d a r y condition. Similarly to the proof of Theorem 5.1, we can show the properties
0 0
==*.
(Ih + I K / 1 L h ) - l f h
_> 0
(6.38)
and
with the well-definedness of (Ih + AIrhlLh) -1 " Xh ~ Xh. On the other hand, Lhl~h is well-defined for X:h 9 Xh with [X:h] - If~-1 fa Xh = 0 by Poincar~-Wirtinger's inequality, similarly to the continuous case. Letting E v = v - [v], we have
IILhlEKhPhlBLI([.~),WI,q(~-~)~ Cq, lim IGL-j1EKsP~v ii i~ h~0
-
L-1Ev
W 1 , q (~"~)
=0
(6.40) (6.4~) (6.42)
220
6. Nonlinear Problems
for q E [1, 2) and v E Ll(f~), corresponding to the results obtained in Theorems 1.12 and 5.4. Inequality (1.48) is also replaced as [[LhTrhL-1E[[L~(a),L~(a) < C, and hence
I Kff I LhzrhL-lzll.(~)...(~)
_< c
(6.43)
follows from (5.7) with p = 2. Property (5.4) also keeps as Uh C X h ~
u
in
LP(f~)
MhUh ~ U in
===:=v
LV(f~).
(6.44)
Wellposedness We set
AhV = I(h 1LhTrhf (v) for v E W = C(ft), which can be regard as an operator in Xh. We have the following. Theorem
6.4. It holds that
IIMh~h[~h -- ~h]+ll~'(~) -- IIM ,. Iv,.-
+ AAhvh- aA,~'Oh]+ll~,(~),
(6.45)
where Vh, Vh E Xh and A > O. Furthermore, (Ih + AAh)Xh = Xh follows. Similarly to the continuous case, the nonlinear semigroup theory guarantees the unique solvability of (6.37) globally in time by Theorem 6.4, because IIMhTrhl'l Ll(ft) provides a norm to Xh. The solution is given as uh(t) = Sh(t)Trhuo for Sh(t) = lira m---,oo
(
Ih + t A b
(6.46)
7Yt
Inequality (6.45) now gives for Vh, 77h C Xh and t _> 0 that
I M,.zrh[S,.(t)v,.- Sh(t)~,,]+llL,(~) < IlMhZr,,[Vh- '&]+IIL'(~)
9
Therefore, u0 >_ 'u0 implies 'uh(t) _> 'Sh(t) for u0, '~),0 E W, where Uh = uh(t) denotes the solution of (6.37) and ~h = ~h(t) that of d d~I(hfih + Lhrchf(ith) = 0
with
/th(O) = 7rhftO.
In particular, u0 > 0 implies uh(t) > O. Furthermore, L 1 stability of the approximate solution holds in the following sense:
IIMh~'h '~'h(t)lllL,(a) 4 Mh~',~luolll,,,(a)" Because {rh} is regular, we have II~,~IXhIIILI(T) ~ Inequality (5.2) with p = 1 can be replaced by
IlxhlIL,(T)
for T E Th and Xh C Xh.
c-' I xh lI L, (T) _< IIMh~h xhlIIs,(T)< c X,,IIL,(T)
(s.47)
6.3. Degenerate Parabolic Equations
221
and hence
II[s,,(t)vh - sh(t)~]+ IIL,(~ ) _< c II[~ - ,~]+ll~,(~)
(6.48)
follows for Vh, 9h C Xh and t _> 0. Those properties are preserved in the time discretized equation, the backward difference finite element approximation
I(h [u;(t + T)~_- uT,(t)] + LhTrhf (u~(t + T ) ) = 0
(6.49)
with u~,(0) = 7rhuo, where t = rnT with m = 0, 1, 2 , . - . . In fact, its solution is given as u[,(t) : ( ~ + ~ A , , ) - m ~ u o for t = m~-, and formula (6.46) reads as
Uh(t) = lira u'h(t ). ~I0
We have
(6.50)
IlM~h bT,(t) - ~T,(t)]+ IIL,(~) _< IIM~,~,~ [~o - ~,o]+1}~,(~) for u0, ~0 E W, where U~h(t) denotes the solution of (6.49) with u~(0) = 7chuo, and 5T,(t) that with ~!~(0) = 7Ch50. We proceed to the proof of Theorem 6.4. Variation and lattice are key structures. Talking about the variational structure, we note that both the solution and test functions are taken from Xh. The lattice structure is also non-trivial, as Uh E Xh does not necessarily imply [Uh]+ E Xh. Given Vh C Xh, we take F~: = {a E Fh I +Vh(a) >_ 0}. It holds that
'V=~~ 7rh [Vh]:l== -nt- E Vh(a)Wa, aeV~h where [ - ] + = max{0, + . }. Then, 0 _ 0 by (6.38). Therefore,
[(Ih Jr-,~KhlLh) -1 Vh]+ 0 holds for v C W = C(f~), where
sgn+ v = ~1 to
(v>O) (v < o).
Proof." Given u, v E W, we have
7rh (u. sgn+v)
~ u(a)wa aeyhn{v>O}
= =
~
u(a)wa = rrh (u. sgn+Trhv).
ae Vhn{ rrhV>_O}
Therefore, f
f
Ja Mhrrh [(K171Lhrrhv) 9sgn+v] : Jn Mh~rh [(Ir
9sgn+vh] ,
where v~ = rrhv E Xh. Writing uh = (Ih + AKt71Lh) -1 vh for A > 0, we have Yh
Uh
-
=
)~ ( I h 2t- / ~ / ( / 7 1 L h ) -1 IV- -h1
Llz~tJh"
This implies A j~ Mhrrh [(Ih+ kKt[1Lh) -1K[[1Lhvh " sgn+vh]
= s Mh~,,[(~,,- ~,,,). ~n+~,,] = s ~,,,~,, [.,,,,]+- s A,,,~,,[.,,,,-~n+~,~] >_ s ~',,~,, [v,,]+- s Mh~,, [~,,,]+ >_0 by (6.51). Letting A I 0, we obtain
.s M,,~,, [( 0. Let vh = (I + AAh)uh and b =
(6.54)
I1~,~11~(~).The relation
(~h, Xh) + A (VTrhf (Uh), VXh) = (Vh, Xh)
(Xh 6 Xh)
(6.55)
holds for Uh = Vh = -+-b E Xh. This means + + ~ h - (Z,~ + ~m,~)-' (+b) = +b and inequality (6.54) is reduced to the following lemma of comparison. consequence of Theorem 6.4 and the proof is complete.
It is a direct []
L e m m a 6.7. Given vh, 9h 9 Xh, let Uh, ith 9 Xh be the solutions of (6.55) and
(~h, Xh) 4- A (VTrhf(~h), VXh) - (-~h,Xh)
(Xh 9 Xh),
respectively. Then, Vh 0 and v C W. A quick overview now guarantees the estimate
II~(t)ll~(~)
_< II~-h~oll~(~)
inductively by #r/o.,- _< 1 and (6.60). We may assume t h a t f = f(u) is Lipschitz continuous with the Lipschitz constant # in R by replacing f(u) for lul >_ 117rhuoIL~(n)if necessary and then
r If(r)-
GT
f(s)l + ( r - s) - L ( f ( r ) GT
f(s)) = I r - sl
(6.61)
follows for r, s E R. Theorem 5.1 assures L1 (f~)
for v C W. Equality (6.61) gives for Xh, Oh C Xh t h a t
IIMhTrh Ifh(T)Xh -- fh(T)~JhlllLl(t2) T < =lMh~hIf (Xh) -- f (~,,)lllL~(~) (7 T
O"m
-
IIMh~,~
Ix~ - r
L 1 (Q)
(6.62)
Convergence (6.58) is a consequence of the nonlinear Chernoff formula. Namely, it suffices to show the convergence of the resolvent in the sense that lim TiO
(
Ih + - ( I h - Fh(r)) 7-
)1
Xh = (Ih +/~Ah) -1Xh,
(6.63)
226
6. Nonlinear Problems
where A > 0 and Xh C X h. Letting Ch = (Ih + )~Ah) -1 Xh, we set ~
r
=
)-1
~,~ + -
(~-
r,~(~))
Xh
T
and
X~=r T
In use of xh = r
+ - (h, - Fh(~))Cj:, T
we have
(
IIMh~h ICh
1+
--
Chlllc,(.) --< IIMhTrh IXh -- x;~lllL,(~) + -IIMhTrh I ~ -- ~hlllLl(a) T
by (6.62). This means
IIMh~hICZ - ~hlllL,(~) ~ IIMh~hI ~ h - ~,SIIIL, 9 Here we have x~r
=
C h _ A-[-
e-aT
K,: I L h T r h f (r
-- zrhf (~Ph)]
Crr
--~ r
+ AK[[1LhTrhf (r
= r
+ AAh~h = Xh
by (6.59). Hence limTl0 ~Pt: = Ch and (6.63) follows. The proof is complete.
C o n v e r g e n c e of R e s o l v e n t Trotter-Kato's theorem assures that convergence of semigroups is a consequence of that of resolvents, and under that spirit, we study the convergence of resolvents now. Here, we take the simplest case that {rh} is uniform" each vertex is shared with 6 triangles and the following property holds in rh, where oz(z) = x + h~ei for a basis {el,e2} of R 2 and uh < h i , h 2 0: T E rh
==*
ai(T) C rh
(i = 1, 2)
Under such a situation, we can show tile following. T h e o r e m 6.8. Under the assumption stated above, it holds t/tat lim II(Ih + AAh) -17rh'v- (1 + AA) -l VllL~(m = 0 hlO
f o r A > O and v E W .
(6.64)
227
6.3. Degenerate Parabolic E q u a t i o n s Proof." Given I > 0 and v E W, we set vh = 7rhv and Uh : (Ih + / ~ A h ) -1 7rhV.
We have I(hUh -+- ALhTChf (Uh) = I(hVh
(6.6s)
II~ll~oo(~) ~ IIv~ll~oo(~)~ II~llLoo(~)-
(6.66)
and (6.54):
Therefore,
III~t~*hllL~(~)< C II~IIL~(~)fonows from (5.6). Similarly, IIKhvhllLoo(a) < C Ilvll~(~)
holds, and we get IILh~hf(uh)llL,(~) 0, we have a measurable set f~ C f~ satisfying If~ \ f~l < s and uh; ~ u uniformly on f~ by Egorov's theorem, where I" I denotes the Lebesgue measure. This implies that 7rhf(Uh,k) ~ Z(U) uniformly on a~ and w = f ( u ) holds in f~. Therefore, (6.67) follows by making s I 0. In this way, 7rh,kf(Uh,k) ~ I(U) weakly in w l ' q ( a ) and Uh,k --~ u strongly in Ll(f~) with some u C L l ( a ) . It holds that
IlL; 1EZ~hUh -- L -1 Eullwl,q(f~) ]IL;1EKhPh (U h - 0 smaller if necessary, lyl < ~ implies
f l ~ ( x + y) - ~(~)1
dx
0 by u0 C W and u[(t) = Ah,~ has the following properties is (2/)~)-Lipschitz continuous and
(I + rAh,~) -1" Xh ---* Xh is a contraction with respect to ]]MhTrh 1" ]]]gl(n)" If t = m r , we have ,u rh,~ (t) --u~(t)
=
(Ih -[- TAh~) , -m 7rh'Uo - ( I + ~-A ~ ) -m Uo
=
(I + TAX)-mTrhUO- (I + "rA,x)-muo
m + E[(I
+ rA~) -(m-e) (Ih + rAh,~)-'
~=1
- - ( I + TA,x) -(m-e+l) (Ih -+- TAh,~)-(e-1)]TrhU0 by the associative law of operators. In use of (6.73), Ll-norm of the second term of the right-hand side is estimated from above by m
Z II[(1 + T A ~ ) ( I h e=l
+ TAh,,x) - e -
(Ih + TAb,A) -(f-l)] 7[hUOIILI(f2)
6.3. Degenerate Parabolic Equations
233
This is equal to
m g=l m g=l
We obtain from (6.73) that
][u~,~(t) -
u~(t)llLl(a) ~
[[(Trh --/h) U0[ILI(~)
m g=l m
(6.81) g=l
In use of the Lipschitz continuity of Aa, the second term of the right-hand side of (6.81) is estimated from above by
2~ m
)-~
-~]
g=l
-< A Ee:l (Ih + TAh,~)-eTrhUo- (I + TA~)-euo +7 ~
(I + ~A~
~o- (I +
Ll(a)
~A~) -~
g--1
m -
X i8 an isomorphism. In particular, we have
c' I ~llv < Ilsl/~Jl~ _< c II,~llv for~e
V.
(7.8)
7. Domain Decomposition Method
246 If h = 7-{~ is suitably regular, we get
f
ob
for r/ E V. This means S'~ = (Oh~On)Iv in the sense of distributions, and in this way the Dirichlet to N e u m a n n map is realized as a self-adjoint operator in L2(',/). We write S = S(t2, 7) and call it the D N map pertaining to (t2, 7), to express this correspondence. The o p e r a t o r S 1/9 can be characterized by the variational principle. In fact, the variational principle for the harmonic equation claims t h a t
IIV~ L=(~)~ IlVv L~(~) for ~ E V, where v E Kl(f~) is so taken as v[~ = ~. This, together with Proposition 7.2, yields the following.
Proposition 7.3. Given~ E V, we have
Is,/= 0 determined only by f~2 such that
I1 ( )11 _< co#(0) I1(~
~
(7.14)
for k = 1 , 2 , 3 , . . . .
Here, we have
f(0)=
1 -0 20-1
(0 < 0 _< 2/3) ( 2 / 3 _ < 0 < 1)
and hence 0 < f(0) < 1 follows if 0 C (0, 1]. Furthermore, (7.14) is equivalent to
I1~~ -
~1~1I~ _< ~o~(0)~1I# ~~ ~1~I1~.
This implies the convergence of the approximate solutions {u}k)}. Theorem
7.9. Under the same assumption of Theorem 7.8, we have
[lUlk ) - ?~ll2,IIHl 0: 7ELTe~I C_ ~2. Then, conditions (Ira) and (I e) imply J l ( { , { ) < rnJ2({,{) and J2({,{) _< gJl({,{), respectively, for { C V. We get the following, where the case g = oo is permitted under the agreement of g-1 = 0. L e m m a 7.13. /f both conditions (fl) and (In) are satisfied, then we have g-1 ~ H < m. 7.14. Under the same assumptions, inequalities (7.14), (7.15), and (7.16) hold with f (0) defined by
Theorem
1_(1_+_e_1)0(0 1, then we have 0 < H < m. If
(R e) is satisfied with g >_ 1 furthermore, then f-1 < H < m follows. Proof: First, we take the case that f~2 is adjacent to 7 from outside, and hence r > R holds on f~2. Let J ' be the J-form pertaining to (~', 7), where s = T,,,,Rfl2. Then we have o"7"1(~,~) J ' ( { , { ) for ~ C V. We take the harmonic extension h.2 C I('(f~2) of ( into s and put v(r', qp) = h2(r, qp) with r' = r + (r - R ) / m . We have v E I ( ' ( ~ ' ) and ,
-
-
L'2(fl,) = .
~
,
k, Or'J
+ 777 ~
(Oh.2 2r' ) -- 9rdrdqp + L
1
(
r'dr'd~
c3h2
)
2
--r 9 l 9rdrdqp
_< mS~(
