Besov regularity for elliptic boundary value problems in polygonal domains

Applied Mathematics PERGAMON

Applied Mathematics Letters 12 (1999) 31-36

Letters

Besov Regularity for Elliptic Boundary Value Problems in Polygonal Domains S. DAHLKE Institut ffir Geometrie und Praktische Mathematik RWTH Aachen, Templergraben 55 52056 Aachen, Germany

(Received and accepted September 1998) Communicated by P. Butzer Abstract--This paper is concerned with the regularity of the solutions to elliptic boundary value problems in polygonal domains f~ contained in R 2. Especially, we consider the specific scale Bar(L.r(f~)), 1/r = ~/2 + 1/p, of Besov spaces. The regularity of the variational solution in these Besov spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. The proofs are based on specific representations of the solutions which were, e.g., derived by Grisvard [1], and on characterizations of Besov spaces by wavelet expansions. (~) 1999 Elsevier Science Ltd. All rights reserved. K e y w o r d s - - E U i p t i c boundary value problems, Adaptive methods, Besov spaces, Wavelets.

1. I N T R O D U C T I O N

Quite recently, the regularity of the solutions to second-order elliptic boundary value problems Lu = f, u = 0,

on ~ C R d, on 0 ~ ,

(1.1)

w h e r e f~ is a L i p s c h i t z d o m a i n , in specific Besov spaces has b e e n i n v e s t i g a t e d , see, e.g., [2,3]. T h e a i m was t o p r o v i d e s o m e t h e o r e t i c a l f o u n d a t i o n s for t h e use of a d a p t i v e schemes for t h e n u m e r i c a l t r e a t m e n t of (1.1). T h i s n o t e c a n b e i n t e r p r e t e d as a c o n t i n u a t i o n of t h e s e studies. T h e o r d e r of c o n v e r g e n c e of u s u a l (linear) G a l e r k i n schemes o b t a i n e d , e.g., b y finite e l e m e n t s p a c e s b a s e d on u n i f o r m g r i d refinement, is d e t e r m i n e d b y t h e r e g u l a r i t y of t h e v a r i a t i o n a l s o l u t i o n u t o (1.1) in t h e u s u a l S o b o l e v scale H a ( f ~ ) , a > 1. U n f o r t u n a t e l y , on a g e n e r a l L i p s c h i t z d o m a i n , t h i s S o b o l e v r e g u l a r i t y m a y n o t be v e r y high, even if t h e r i g h t - h a n d side f is sufficiently s m o o t h . T h i s fact is c a u s e d b y s i n g u l a r i t i e s n e a r t h e b o u n d a r y . Therefore, t o increase efficiency, one often uses adaptive m e t h o d s , i.e., t h e u n d e r l y i n g g r i d is o n l y refined in regions w h e r e t h e s o l u t i o n lacks s m o o t h n e s s . I n t h i s case, one does n o t use t h e w h o l e linear spaces, hence, a n a d a p t i v e s c h e m e c a n This work has been supported by Deutsche Forschungsgemeinschaft (DA 117/13-1). The author takes advantage of this publication to warmly thank A. Cohen for numerous inspiring discussions and lots of helpful remarks. He also feels grateful to the referees for several valuable comments. 0893-9659/99/$ - see front matter. (~) 1999 Elsevier Science Ltd. All rights reserved. PII: S0893-9659(99)00075-0

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32

S. DAHLKE

be interpreted as some kind of nonlinear approximation. Then the question arises if nonlinear methods indeed provide some gain of efficiency when compared with linear schemes. We shall restrict ourselves in the sequel to the specific setting of nonlinear wavelet approximation since the theory is almost fully developed in this case (although similar results hold in other settings, see [4,5]). Let k9 be the set of 2 d - 1 functions built in the usual way by tensor products from the univariate orthonormal Danbechies wavelets, see [6,7]. Then the functions

~7I : : ~Tj,k := 2Jd/2~7(2J " --k),

I = 2-Jk + 2-J[O, 1]d,

k E zd, j E Z, ~Te ~P,

(1.2)

form an orthonormal basis for L2(Rd). In nonlinear wavelet approximation, we approximate a function F E Lp(]~ d) by the nonlinear manifolds Adn of all functions

G :

~

ai,~?r]l,

(z,n)er with F c 7) x ~ of cardinality n, where 7) denotes the set of all dyadic cubes in R u. Then we set

an(F)L~(R~) :=

inf I I F - GIIL~,(Rd). GE~4n

(1.3)

In this case, one has the following characterization (see [4]) for 1 < p < c~,

1 < o0 if and only if F E B~(L~(Rd)), Eoo [n(~/U(Tn(F)L~(~) IT --n

T=

(d

+

~)--1 ,

(1.4)

n~--I

where the B~(Lr(Rd)) are the Besov spaces (see, e.g., [8,9] for the definition and the main properties of Besov spaces). Having adaptive schemes in mind, these results on nonlinear wavelet approximation make it natural to ask the following question: what is the regularity of solution u to (1.1) as measured in the scale B ~ ( L r ( ~ ) ) , T = (a/d + l / p ) - 1 ? Especially, does the solution have a higher smoothness order in these spaces compared to the usual Sobolev scale? For then, adaptive methods on nonsmooth domains can perform better than linear schemes, in principle. The results in [2,3] indicate that this is indeed the case for many problems. However, all these investigations were concerned with general Lipschitz domains, i.e., all boundary points are viewed as equally 'bad' which is often not realistic. In practice, one is typically concerned with domains with piecewise analytic boundary, e.g., with polyhedral domains. One would expect that in this case much sharper results are available. The results of this paper show that this is indeed the case, at least for specific problems in ]t 2. Surprisingly, it turns out that the Besov regularity of u is independent of the shape of the domain and is only determined by the smoothness of the right-hand side (which is clearly false for the usual Sobolev scale). 2. M A I N

RESULTS

In this section, we want to state and to discuss the main results of this paper. They rely on the fundamental Theorem 2.3 which will be proved in Section 3. In the sequel, ~ will always denote a simply connected polygonal domain in R 2. The segments of 0 ~ are denoted by Ft, F~ open, l --- 1 , . . . , N, numbered in positive orientation. Furthermore, St denotes the endpoint of Ft and wl denotes the measure of the interior angle at Sl. We consider the model problem Au = f, on ~, u = 0, on 012. (2.1) We want to investigate the dependence of the Besov regularity of the weak solution u to (2.1) on the smoothness of the right-hand side f and on the shape of the domain. The following result shows that for the Besov scale B~(L~(~)), 1 / r = c~/2 + 1/2, the influence of the domain is neglectable. (We shall primarily be concerned with this specific scale since we are mainly interested in approximation with respect to L2(~).)

Besov Regularity

33

THEOREM 2.1. Suppose that the right-hand side f in (2.1) is contained in HS(f~) for some s _> - 1 . Furthermore, let us assume that mTr/w~ ~ s + 1 for ali l = 1 , . . . , N, m >_ 1. Then solution u to (2.1) satisfies

uEB~(L~(f~)),

0
1

(~+1~ \z z/

T

(2.2)

PROOF. The proof is based on the fact that u can be decomposed into a regular part un and a singular part us, u = un + us, where un E HS+2(f~) and us only depends on the shape of the domain and can be computed explicitely. Results of this form were first derived by Kondrat'ev [10]. In this note, our standard reference will always be the book of Grisvard [1]. We introduce polar coordinates (rl, 0l) in the vicinity of each vertex St and introduce the functions

~ql,m(rl, Of) = (l (rl)r~l ''"~ sin

( m~rO, ~ mlr , when Al,m := is not an integer, \ wt / wl

(2.3)

Sz,m(ri,01) = Q(rz)r~"" [logrl sin (mlrOl ~ + Oleos (mrOz ~] otherwise, \ wz ] \ wl / j where (t denotes a suitable C °o truncation function. Then one has the following theorem (see, e.g., [1, Chapter 2.7]). THEOREM 2.2. For given f E HS(f~), s > - 1 , the corresponding variational solution to (2.1) has an expansion u = un + us, where uR E HS+2(~) and N

=E

Z

(2.,)

j = l 0
provided that no Al,m is e q u a / t o s + 1. According to Theorem 2.2, we have to establish Besov regularity for both parts un and us. By the embeddings of Besov spaces: H ' + 2 ( ~ ) = B~+2(L2(f~)) ~ B~+2(Lr(f~)) ¢-~ B~(L~(~2)), we have un E B~(Lr(f~)) for any a , r as in the statement of Theorem 2.1. It remains to study the singular part us. It turns out that this part,' although not very smooth in the usual Sobolev scale, has arbitrary high smoothness in the specific Besov scale we are interested in. THEOREM 2.3. A n y function St,,~ defined by (2.3) satisfies

St,m 6 Bra(Lr(a)),

for all a > 0,

1 a 1 + • T 2 2

(2.5)

By employing Theorem 2.3, the result follows. II Theorem 2.3 can also be used to derive further results concerning nonlinear approximation with respect to Sobolev norms. In the context of symmetric second-order elliptic problems, these norms axe in a certain sense more natural since the Galerkin method provides quasi-optimal schemes with respect to H 1. It is well known that for nonlinear wavelet approximation with respect to H t, a characterization similar to (1.4) holds, but with the scale B~ (Lr (Rd)), 1 / r = a / d + 1/p replaced by Bar+t(Lr(Rd)), 1/~" = a / d + 1/2, see, e.g., [11,12]. Concerning the regularity in these spaces, the following holds. THEOREM 2.4. Suppose that the conditions of Theorem 2.1 are fulfilled with s > - 1 / 2 . Then solution u to (2.1) satisfies

u E B~+a/2(Lr(~)),

1

O < a < S + z,

1

T

PROOF. From (2.3), it is easy to check that St,m E H3/2+e(f~) = B3/2+'(L2(~)) for some sufficiently small e > 0 depending only on the shape of the domain. Therefore, by interpolation between Ha/2+e(f/) and Ba~(L~(fl) ), 1Iv = a/2 + 1/2, one obtains that St,m e Ba~/2+a(L~(fl) ), 1/r = a / 2 + 1/2, for all a > 0. Again by using embeddings of Besov spaces, the result follows. II

34

S. DAHLKE

3. PROOF

OF

THEOREM

2.3

We want to establish Besov regularity for each of the functions St,re. Therefore, we fix the parameters l and m and use the abbreviations r = rt, 0 = 01, 7 = )~t,m, and S = St,re. We shall only consider the case that 7 is not an integer, the other case can be studied analogously. The proof is based on wavelet analysis. We may use the fact that Besov spaces can be characterized by wavelet expansions. If the functions ~ • kv are sufficiently smooth (which can always be achieved, see [6] for details), then a function F is in the Besov space Bar(L~(Rd)), 1/T = (~/d+ 1/2, if and only if

IIP°(F)IIL.(Ra) +

Z

Z

I(F'~/x)I~

< co,

(3.1)

~E~ IE2~+

where :D+ denotes the set of all dyadic cubes of measure < 1 and Po is a projector onto a suitable subspace of L2(Rd), see, e.g., [7] for the case T > 1 and [13] for the general case. To use this characterization, the first step is to extend S in a suitable way to all of R 2 as follows. We may assume that • consists of compactly supported functions, see again [6], so that supp(~/1) is contained in a cube Q(I), IQ(I)I 5 III• (In this paper, ' ~< ' indicates inequality up to constant factors.) We introduce the distance to the vertex St, 61:=

inf r(x). xeQ(r)

(3.2)

Furthermore, we set U := fl M B(St, R), where supp (t C [ - R , R] and B(SI, R) clearly denotes the ball of radius R at St. From (2.3), we observe that S has a trivial extension onto a cone V centered at St, containing U, and chosen such that for some suitable constant C ,

Q(I) c V

if

Q(I) M U

# @,III = 2 -25

and

~I > C2 - j -

(3.3)

We denote this new function also by S. It is at least contained in H3/2(V) and we may use a Whitney extension to obtain a function on all of R 2 for which we again keep the notation S. Then, on the old domain fl, S has an expression

S = P0(S) + Z

(S,~/I)~/I,

(3.4)

(L,7)eh where A denotes the set of all pairs (I,~), I • 7) +, ~ E 9 , for which Q(I) n U # O. It can be checked that in our case, P0(S) is 'harmless', see [2]. Therefore, it remains to treat the second term in (3.4). Let us start by estimating one wavelet coefficient. By using the fact t h a t each Yx has a certain number of vanishing moments according to its smoothness, see again [6] for details, and employing a classical Whitney-type estimate for the error of approximation by polynomials on cubes, it turns out that there exists a polynomial PI of total degree < n such that

I1

IIS PIIIL~(Q(x))IITIlHL2(Q(I))~ IQ(I)I('~+I)/2I'NIw"(L=(Q(1))) -

2-j(n+l)

ISIw,~(L,,~(Q(I))).

(3.5)

Now we have to sum these expressions. First, we fix a refinement level j by considering the set

At :=

(3.6)

• A I I/I = 2-2 } •

For each refinement level, we cover the subdomain U by layers, i.e., we define

hj,k := {(I, 7) • At I k2-

<

< (k + 1)2-

}.

(3.7)

BesovRegularity

35

The wavelets in the vicinity of St have to be treated separately. Therefore, we first deal with the

sets

I 5i < C2-J}.

A~ := As \As,c, Aj,c := {(I,~) • Aj

(3.8)

Since

IAs,kl <~ k

(3.9)

and

ISIWn(L~(Q(I)) ) ~,~517-n,

for

(I, 7) • h~,

(3.10)

we obtain co

oo

E E 2--J(nT1)'r~7--n)T ~'~ E k" 2-J(n"{-1)V(k-2-J)(?-n)'r E IVr ~ k=kl (I,~)EA~ (I,~?)EAj,k k=kl oo

(3.11)

< 2-~(~+~)r ~ kl+(~-nlL k=kl

where kl depends on the constant C in (3.3). If we choose n large enough, the sum involving k is clearly finite. Summing over all refinement levels, we are left with a geometric series which is clearly convergent. It remains to study the sets Aj,e. Using the fact that S E H3/2(R2), this can be performed by following the lines of the proof of Theorem 3.2 in [2]. We obtain the condition 3r -~<0, 2-~which is clearly satisfied. The theorem is proved.

|

REMARK 3.1. In principle, Theorem 2.3 can also be proved by using directly the definition of the Besov spaces, i.e., by estimating the modulus of smoothness for the singularity functions according to (2.3), etc. However, to our impression, the method proposed here is considerably simpler from the technical point of view. Moreover, this kind of proof fits very naturally into the setting of nonlinear wavelet approximation we are interested in. Let us also mention t h a t Besov regularity results can be obtained by studying the convergence rates of certain grading strategies, see [12] for details.

4. F U R T H E R

RESULTS

Finally, we want to state some further results without proofs for other interesting special cases where suitable decompositions into regular and singular parts are available. First of all, let us mention the following observation concerning regularity and approximation with respect to Lp(~), 1 < p < oo. THEOREM 4.1. Suppose that the right-hand side f in (2.1) is contained in W ~ ( L p ( ~ ) ) , 1 < p < oo, ~ > 0 an integer, and suppose that no At,m is equal to ~ + 2 - 2/p. Then the variational solution u to (2.1) satisfies u6B~(Lr(~)),

J+2>a>0,

1

T

-

c~

2

+

I

p

•

(4.1)

We are also able to treat the biharmonic equation A2u = f,

on ~,

~U

u = O n t = 0,

where clearly nt denotes the outward normal at Ft.

on Ft, l = 1 , . . . N ,

(4.2)

36

S. DAHLKE

THEOREM 4.2. L e t us suppose that t h e right-hand side f in (4.2) is contained in H - l ( f ~ ) and let us f u r t h e r m o r e a s s u m e that f~ has no v e r t e x St with wt = tan(wl). T h e n the variational solution u

to (4.2) satisfies ueB~(Lr(~)),

1

0
r

--

a 1 + 2 2

(4.3)

F i n a l l y , let us m e n t i o n t h e e l a s t i c i t y s y s t e m in 2D. G i v e n f E L2(f~) 2, g E L 2 ( 0 ~ ) 2, one considers the problem 2

ZDjaij(g)+fi----O, j=l

2

i = 1,2 o n e ,

Za,,j(g)nj=g,, j=l

i = 1, 2 on 0 ~ ,

(4.4)

w h e r e ~(zT) d e n o t e s t h e stress tensor. I t c a n b e shown t h a t u n d e r s o m e n a t u r a l c o n d i t i o n s ~7 e B r a ( L r ( ~ ) ) 2, 0 < ~ < 2, 1/7" = c~/2 + 1/2.

REFERENCES 1. P. Grisvard, Singularities in Boundary Value Problems, Research Notes in Applied Mathematics, Vol. 22, Springer, Berlin, (1992). 2. S. Dahlke and R. DeVore, Besov regularity for elliptic boundary value problems, Comm. Partial Differential Equations 22 (1/2), 1-16, (1997). 3. S. Dahlke, Wavelets: Construction Principles and Applications to the Numerical Treatment of Operator Equations, Shaker, Aachen, (1997). 4. R. DeVore, B. Jawerth and V. Popov, Compression of wavelet decompositions, Amer. J. Math. 114, 737-785, (1992). 5. R. DeVore and X.M. Yu, Degree of adaptive approximation, Math. Comp. 55, 625-635, (1990). 6. I. Daubechics, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Math., Vol. 61, SIAM, Philadelphia, (1992). 7. Y. Meyer, Wavelets and Operators, Cambridge Studies in Advanced Mathematics, Vol. 37, Cambridge, (1992). 8. R. DeVore and V. Popov, Interpolation of Besov spaces, Trans. Amer. Math. Soc. 305, 397-414, (1988). 9. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, (1978). 10. V.A. Kondrat'ev, Boundary-value problems for elliptic equations in domains with conical or angular points, Trans. Moscow. Math. Soc. 16, 227-313, (1967); Translated from: Tr. Mosk. Mat. Obshch. 16, 209-292, (1967). 11. S. Dahlke, W. Dahmen and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, In Multiscale Wavelet Methods for Partial Differential Equations, (Edited by W. Dahmen et al.), pp. 237-283, Academic Press, San Diego, CA, (1997). 12. R. Hochmuth, Restricted nonlinear approximation and singular solutions of boundary integral equations, Preprint Nr. A/13/98, Free University of Berlin, (1998). 13. M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34, 777-799, (1985).