Parallel Galerkin Domain Decomposition Procedures for Parabolic Equation on General Domain

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1 Parallel Galerkin Domain Decomposition Procedures for Parabolic Equation on General Domain Keying Ma, 1 Tongjun Sun, 1 Danping Yang 1 School of Mathematics, Shandong University, Jinan 50100, People s Republic of China Department of Mathematics, East China Normal University, Shanghai 0006, People s Republic of China Received 13 December 007; accepted 4 August 008 Published online 5 November 008 in Wiley InterScience ( DOI /num.0394 Parallel Galerkin domain decomposition procedures for parabolic equation on general domain are given. These procedures use implicit Galerkin method in the subdomains and simple explicit flux calculation on the interdomain boundaries by integral mean method or extrapolation method to predict the inner-boundary conditions. Thus, the parallelism can be achieved by these procedures. These procedures are conservative both in the subdomains and across interboundaries. The explicit nature of the flux prediction induces a time-step limitation that is necessary to preserve stability, but this constraint is less severe than that for a fully explicit method. L -norm error estimates are derived for these procedures. Compared with the work of Dawson and Dupont [Math Comp 58 (199), 1 35], these L -norm error estimates avoid the loss of H 1/ factor. Experimental results are presented to confirm the theoretical results. 008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 5: , 009 Keywords: domain decomposition procedures; extrapolation method; Galerkin method; integral mean method; parabolic equation I. INTRODUCTION It is well known that most practical problems in engineering can be turned into solving large-scale partial differential equations (PDEs). Parallel algorithms, based upon overlapping or nonoverlapping domain decompositions, are effective ways to solve the large scale of PDE systems (for examples, see [1 15]). In [6], Dawson and Dupont applied explicit/implicit conservative Galerkin domain decomposition procedures for parabolic equations. They used implicit Galerkin procedures in the subdomains Correspondence to: Keying Ma, School of Mathematics, Shandong University, Jinan 50100, People s Republic of China ( makeying@sdu.edu.cn) Contract grant sponsor: National Basic Research Program of P. R. China; contract grant number: 005CB31703 Contract grant sponsor: China State Education Ministry 008 Wiley Periodicals, Inc.

2 1168 MA, SUN, AND YANG and explicit calculations on the interdomain boundary to predict the flux by use of some weight functions approximating to the delta functions. These procedures were conservative both in the subdomains and across interdomain boundaries. The explicit nature of the flux prediction induced a time-step limitation that was necessary to preserve stability, but less severe than that of a fully explicit method. They derived a priori error bounds in terms of the errors of certain elliptic approximations rather than powers of some asymptotic parameter. But, in fact, there was a loss of H 1/ factor for space variable. It was noted that the loss could be avoided in certain special cases using some techniques of [16] but was not improved in [6]. Dawson Dupont s schemes were designed for PDEs of special form such as (a u) and special domain such as rectangle-type domain. So there are some difficulties to extend these schemes to problem of general form on general domains. In this article, we present parallel Galerkin domain decomposition procedures for parabolic equation of general form on general domain. These procedures are also explicit/implicit schemes. They use implicit Galerkin methods in the subdomains and other explicit calculations to predict the flux on. First, an integral mean method is utilized to present a simple explicit flux calculation. This calculation is just the integral mean value of the directional derivative of the solution on over a strip domain with width H. We call this scheme as integral mean parallel Galerkin scheme (IM-PG). Second, to improve higher order accuracy with respect to H, we extrapolate the flux calculation and derive another scheme. We call this case as extrapolation-integral mean parallel Galerkin Scheme (EIM-PG). Some constraints for time step are still needed for these procedures to preserve stability, but less severe than that for fully explicit methods. With respect to the accuracy order of h, L -norm error estimates are optimal for higher-order finite element spaces (r ) and almost optimal for linear finite element space (r = 1) in two-dimensional domain. Compared with [6], these L -norm error estimates avoid the loss of H 1/ factor. This article is organized as follows. In Section II, we present IM-PG scheme and EIM-PG scheme on a general domain, respectively. In Sections III and IV, L -norm error estimates are derived for IM-PG scheme and EIM-PG scheme, respectively. In Section V we present results of some numerical experiments to confirm theoretical results. Finally, the last section describes the conclusion and perspective. Throughout the analysis, the symbols C, C 1, C 1, K,..., etc., will denote generic constants independent of mesh parameters t and h. Constants K, C are not necessarily the same at different occurrences. The symbol ε will denote smaller positive constants. II. DOMAIN DECOMPOSITION PROCEDURES Let be a spatial domain in R d (1 d 3) with a piecewise uniformly smooth Lipschitz boundary. Let L p ( ),1 p, denote the standard Banach spaces, with denoting the L norm and p for the L p norm. Denote by H m ( ) and Wp m ( ) the standard Sobolev spaces on, with norms m and m,p, respectively. For a time interval [α, β] [0, T ] and a normed space X = X( ), we use the notation L p (α,β;x) to denote the norm of X-valued functions f with the map t f(, t) X belonging to L p (α, β). We consider the following parabolic initial-boundary value equation: u (A u) + cu = f, in (0, T ], t (A u) ν = 0, on (0, T ], u = u 0, in, t = 0, (.1)

3 DDM FOR PARABOLIC EQUATION ON GENERAL DOMAIN 1169 FIG. 1. The domain with the interdomain boundary. where A = (a ij (x)) d d is a uniformly positive definite matrix function, i.e., there exist positive constants 0 <a 0 a 1 such that a 0 d i=1 ξ i d a ij (x)ξ i ξ j a 1 i,j=1 d i=1 ξ i, ξ Rd, x, ν is the unit vector outward normal to, c = c(x, t), u 0 = u 0 (x), and f = f(x, t) are given functions. We begin to consider the Galerkin domain decomposition procedures. Without losing generality, we only discuss the case of two subdomains. But the algorithms and theories can be extended to the case of many subdomains. Divide into two subdomains i for i = 1, by an interdomain boundary as Fig. 1. is a surface of dimension d 1. We denote by i = i the part of the boundary of the subdomains which coincides with. Let ν denote the unit vector normal to, which points from 1 toward. Let T h i be quasi-uniform partitions of i for i = 1orandT h = T h 1 T h, where h denotes the maximal element diameter. We construct the finite element space M h on T h which satisfies the following condition (I): 1. For j = 1,, let M h j be a finite element subspace of H 1 ( j ), and let M h L ( ) such that if v M h, then v j M h j.. For j = 1,, P r ( j ) M h j, where P r( j ) is a polynomial space of degree at most r. 3. For j = 1,, h (0, 1], some integer k 1 and u H k ( j ), there exists a positive constant K independent of h such that where σ = min(r + 1 s, k s). inf u v H s v M h ( j ) Ch σ u H k ( j ), 0 s 1, (.) j From the definition above, we note that functions v in M h have a well-defined jump [v] on : [v](x) = v(x + ) v(x ), x on,

4 1170 MA, SUN, AND YANG FIG.. The strip domain G with width H. where v(x ± ) = lim t 0 ± v(x + tν ). To construct parallel algorithm, for small constant H>0, we introduce an integral mean value of a given function W L ( ) on the inter domain boundary as W H (x) = 1 H W(x + tν )dt, x on. (.3) H H Furthermore, we define the extrapolation of W H (x) on as Ŵ E (x) = 4W H/(x) W H (x), x on. (.4) 3 Generally, near the intersection of boundary and inner boundary, the value of W outside is needed for W and Ŵ E. Let x denote the symmetric point of x / with respect to. For a given function u L ( ), we define { u(x), if x, Eu(x) = u( x), if x /. (.5) By (.5), we know W and Ŵ E have the values on a strip domain G ={y y = x+tν, t [ H, H ], x on } (see Fig. ). Let t be time-step size, integer N = T / t, t n = n t, t U n = (U n U n 1 )/ t, n = 1,..., N. We take the bilinear form D(φ, ψ) = (A φ, ψ) + (cφ, ψ), φ, ψ H 1 ( ). We define two parallel Galerkin schemes as follows.

5 DDM FOR PARABOLIC EQUATION ON GENERAL DOMAIN 1171 A. Integral Mean Parallel Galerkin Scheme (IM-PG) Given an initial function U 0 M h, seek {U n } N n=1 Mh such that where ( t U n, v) + D(U n, v) + ( U n 1 µ,h, [v]) + (v µ,h, [U n 1 ]) + a 1 KH 1 ([U n 1 ], [v]) = (f n, v), v M h, (.6) { 1, if G, U n 1 µ = (A EU n 1 ) ν, K =, if G. (.7) The choice of K will be explained in Lemma 3.. To get higher order accuracy with respect to H, we define another scheme. B. Extrapolation-Integral Mean Parallel Galerkin Scheme (EIM-PG) Given an initial function U 0 M h, seek {U n } N n=1 Mh such that ( t U n, v) + D(U n, v) + ( Û n 1 µ,e, [v]) + ( v µ,e, [U n 1 ] ) + a 1 KH 1 ([U n 1 ], [v]) = (f n, v), v M h, (.8) where U n 1 µ and K are defined as same as (.7). In the scheme (.6) and (.8), the flux on is computed explicitly from U n 1, so that U n can be computed on 1 and fully parallel once U n 1 has been got. These schemes are conservative in the same sense as that mentioned in [6]. Here we first state main convergence results of these schemes, whose proofs will be given in Sections III and IV, respectively. Theorem.1. Let u and {U n } N n=1 be the solutions of parabolic equation (.1) and IM-PG scheme, respectively. Suppose that u is sufficiently smooth and U 0 M h is taken to be W 0, which is defined by (3.1). For linear finite element spaces in two-dimensional domain, let H = O((1+ ln h ) ). Then there exists a constant C independent of mesh sizes h, H and t, such that provided where C 1 is given by (3.34). max 1 n N un U n L ( ) C{ t + h r+1 + H 5/ }, (.9) t C 1 H, (.10) Theorem.. Let u and {U n } N n=1 be the solutions of parabolic equation (.1) and EIM-PG scheme, respectively. Suppose that u is sufficiently smooth and U 0 M h is taken to be W 0, which is defined by (4.11). For linear finite element spaces in two-dimensional domain, let H = O((1 + ln h ) ). Then there exists a constant C independent of mesh sizes h, H and t, such that max 1 n N un U n L ( ) C{ t + h r+1 + H 9/ }, (.11)

6 117 MA, SUN, AND YANG provided t C 1 H, (.1) where C 1 is given by (4.16). Remark 1. For linear finite element spaces, i.e., r = 1, the convergence theorems show that a weaker restraint of mesh ratio between t and h than that of explicit Galerkin schemes. In explicit Galerkin schemes, mesh ratio t/h C is necessary, where some constant C > 0. But in our IM-PG scheme, if we take H 5/ = O(h ) to balance error accuracy with respect to h and H, only mesh ratio t/h 8/5 C 1 is required for some constant C 1 > 0. It is easy to see that mesh ratio of IM-PG scheme is h /5 multiple of that of explicit Galerkin schemes. For the case of linear finite element (r = 1), the condition H = O((1 + ln h ) ) is only necessary in theoretical analysis for the convergence. Analogically for EIM-PG scheme, if we take H 9/ = O(h ) to balance error accuracy with respect to h and H, only the condition t/h 8/9 C 1 is needed for some constant C 1 > 0. The mesh ratio of EIM-PG scheme is h 10/9 multiple of that of explicit Galerkin schemes. With respect to the accuracy order of h, we know that (.9) and (.11) are optimal for higherorder finite element spaces (r ) and almost optimal for linear finite element space (r = 1) in two-dimensional domain. III. CONVERGENCE ANALYSIS OF THE IM-PG In this section, we will prove Theorem.1. We need some following lemmas. Lemma 3.1. For smooth enough function W, there hold estimates W H W L () H W L ( ) (3.1) and and W H W L () CH W W, ( ) (3.) W(x) W H (x) = 1 6 H W ν (x) 1 10 H 4 W ν 4 (x) + o(h 6 ), x on, (3.3) where W ν and W ν 4 are the second and fourth normal derivative of W on, respectively. Proof. By using Taylor expansions, we see that W H (x) W(x) = 1 H t W t (x + t ν )dt dt, H H 0 x on and W H (x) W(x) which implies (3.1). H W t (x + tν ) dt [ H 1/ H W(x + tν ) dt], H H

7 Furthermore, we see that DDM FOR PARABOLIC EQUATION ON GENERAL DOMAIN 1173 W(x) W H (x) = 1 4H H H [ Wν (x + tν )t + o(t 3 ) ] dt = 1 6 H W ν (x) 1 10 H 4 W ν 4 (x) + o(h 6 ), x on, (3.4) which leads to (3.) and (3.3). Lemma 3.. small, we have Let G ={y y = x + tν, t [ H, H ], x on }. Ifψ H 1 ( ) and H>0 is Eψ L (G) K ψ L ( ), (Eψ) ν L (G) K ψ L ( ), (3.5) where K = { 1, if G,, if G. (3.6) Proof. It is obvious that (3.5) holds when G, so we only consider G case. Let G = G, and G be the counterpart domain of G by the definition (.5). That is, for a given point x G, we have a symmetric point x G. For small H>0, we have Eψ(x) L (G) Eψ(x) L ( ) + Eψ(x) L (G ) ψ(x) L ( ) + ψ( x) L (G ) For x = (x 1,..., x d ),wehave (ψ(x)) ν = ψ(x) x 1 cos ( x 1, ν ) + ψ(x) x ψ(x) L ( ) + ψ(x) L ( ) ψ(x) L ( ). (3.7) cos ( x, ν ) + + ψ(x) x d cos ( x d, ν ), (3.8) where cos ( x 1, ν ) + cos ( x, ν ) + +cos ( x d, ν ) = 1. By Cauchy s inequality, we can get ( ) ψ(x) ( ) ψ(x) ( ) ψ(x) ( (ψ(x)) ν ) + + +, (3.9) x 1 x x d and Hence by (3.7) and (3.10), we obtain (ψ(x)) ν L ( ) (ψ(x)) L ( ). (3.10) (Eψ(x)) ν L (G) (Eψ(x)) ν L ( ) + (Eψ(x)) ν L (G ) ψ(x) L ( ) + ψ( x) L (G ) ψ(x) L ( ) + ψ(x) L ( ) ψ(x) L ( ). (3.11)

8 1174 MA, SUN, AND YANG From (3.7) and (3.11), we complete the proof. To obtain error estimates, we introduce an elliptic projection W M h of the solution u as follows: (A (u(, t) W(, t)), v) + (c(u(, t) W(, t)), v) = 0, v M h. (3.1) It is clear that the auxiliary problem (3.1) is equivalent to ( (A W, v) + (cw, v) = f u ) t, v + ( 1) j+1 (g, v), v M h j, j = 1,, (3.13) where g = (A u) ν. These are two standard finite element equations. Let η = u W. From [17 0], we see Lemma 3.3. For η defined by (3.1), there holds L -norm error estimate L -norm error estimate max 0 t T η L ( ) + η t L (0,T ;L ( )) Ch r+1 { u L (0,T ;H r+1 ( )) + u t L (0,T ;H r+1 ( ))}, (3.14) η L ( ) + η t L ( ) Ch lnh { u W, ( ) + u t W, ( )}, if r = 1, d =, (3.15) η L ( ) + η t L ( ) Ch r+1 { u W r+1, ( ) + u t W r+1, ( )}, if r>1. (3.16) For functions ψ with restrictions in H 1 ( 1 ) H 1 ( ), we define a norm ψ = (A ψ, ψ) + a 1 KH 1 ([ψ], [ψ]). (3.17) Lemma 3.4. There exists a positive constant C 0 = 1 such that for small H>0, C 0 ψ (A ψ, ψ) + a 1 KH 1 ([ψ], [ψ]) + (ψ µ,h, [ψ]), ψ M h. (3.18) Proof. By (3.5), we have H ψ µ,h L = 1 () 4H = 1 [ H 4H H 1 4H [ H H a 1 (A Eψ, Eψ) G [ H A Eψ(x + tν ) ν dt] d H ( ν T A1/ ν )( ν T A1/ Eψ ) (x + tν ) dt] d ( Eψ A Eψ)(x + tν ) 1/ ν Aν 1/ dt] d a 1K (A ψ, ψ). (3.19)

9 DDM FOR PARABOLIC EQUATION ON GENERAL DOMAIN 1175 Then, we can derive (A ψ, ψ) + a 1 KH 1 ([ψ], [ψ]) + (ψ µ,h, [ψ]) [ (A ψ, ψ) + a 1 KH 1 [ψ] L ε(a ψ, ψ) + a ] 1K () ε H 1 [ψ] L () ( (1 ε)(a ψ, ψ) ) a 1 KH 1 [ψ] L ε. (3.0) () Here, we used the ε-inequality αβ εα + 1 4ε β, α, β>0 and ε is a smaller arbitrary positive constant. This inequality will be applied frequently in the following analysis. Taking ε = in (3.0) leads to (3.18). The proof of Lemma 3.4 ends. As we have shown, the IM-PG scheme (.6) includes two terms on the interdomain boundary by integral mean method to present explicit flux calculation. These terms are distinct ones different from Dawson Dupont s schemes such that the standard elliptic projection (3.1) is insufficient for optimal error estimates. To get optimal error estimates, we need a new elliptic projection including terms on. This new elliptic projection W M h of the solution u is defined as: (A (u W), v) + ( (u W) µ,h, [v] ) + ( v µ,h, [u W] ) + a 1 KH 1 ([u W ], [v]) = 0, v M h. (3.1) It follows from Lemma 3.4 that the projection problem (3.1) has unique solution for small H. Let The following lemma gives the bounds of ξ n. ξ n = u n W n, θ n = U n W n. (3.) Lemma 3.5. There hold a priori estimates: ξ L ( ) C{h r+1 + H 1/ η L ( )} (3.3) and ξ t L ( ) C{h r+1 + H 1/ η t L ( )}. (3.4) Proof. First, we prove ξ Ch r u r+1. (3.5)

10 1176 MA, SUN, AND YANG From Lemma 3.4 and (3.1), we know C 0 W v (A ( W v), ( W v)) + a 1 KH 1 ([ W v], [ W v]) + ( ( W v) µ, H, [ W v] ) (A (u v), ( W v)) + a 1 KH 1 ([u v], [ W v]) + ((u v) µ, H, [ W v]) + ( ( W v) µ, H, [u v] ) C u v W v, v M h. By trace theorem, we further have inf W v C inf u v v M h v Mh { C inf (u v) v M h 1, A + a 1KH 1 [u v] L () where v 1,A = (A v, v). Hence, we get C { h r u r+1 + H 1 h r+1 u r+1 } 1/ } 1/ Ch r u r+1, (3.6) ξ inf W v + inf u v Chr u r+1. v M h v M h This implies (3.5). Second, we prove such that W W C { H 1 η L ( ) + η } (3.7) W W L () C { η L ( ) + H 1/ η }. (3.8) By (3.1) and (3.1), we know that for each v M h (A (W W), v) + a 1 KH 1 ([W W ], [v]) + ( W µ,h W µ,h, [v] ) + ( v µ,h, [W W ] ) = (A (W u), v) + a 1 KH 1 ([W u], [v]) + (W µ,h ū µ,h, [v]) + (v µ,h, [W u]) = a 1 KH 1 ([W u], [v]) + (W µ,h ū µ,h, [v]) + (v µ,h, [W u]) (c(w u), v). Taking v = W W and using Lemma 3.4, we have C 0 W W a 1 KH 1 ([W u], [W W ]) + (W µ,h ū µ,h, [W W]) + ( W µ,h W µ,h, [W u] ) (c(w u), W W) a 1 KH 1 [W u] L () [W W ] L () + CH 1 W u L ( ) [W W ] L () + CH 1/ W u L ( ) W W 1 + C u W W W,

11 where we used the following inequality DDM FOR PARABOLIC EQUATION ON GENERAL DOMAIN 1177 ϕ µ,h L () CH 1 ϕ L (G), ϕ M h, which is derived straightly from (.3). Hence, (3.7) is derived. Third, we consider L -norm error estimate of ξ. Introduce an auxiliary function ω H 1 ( ) such that { (A ω) + cω = ξ, in, (3.9) (A ω) ν = 0, on. It is well known that ω H ( ) and satisfies By using (3.1), we have ω C ξ. (3.30) ξ = (A ω, ξ)+ (ω µ, [ξ]) + a 1 KH 1 ([ω], [ξ]) + (ξ µ,h, [ω]) = (A (ω I h ω), ξ)+ a 1 KH 1 ([ω I h ω], [ξ]) + (ξ µ,h, [ω I h ω]) + (ω µ,h I h ω µ,h, [ξ]) + (ω µ ω µ,h, [ξ]), (3.31) where ω µ = (A ω) ν and I h ω is the Lagrange interpolation of ω in M h. We turn to bound the terms on the right-hand side of (3.31). By interpolation theorem and (3.5), it is clear that (A (ω I h ω), ξ)+a 1 KH 1 ([ω I h ω], [ξ]) +(ξ µ,h, [ω I h ω]) +(ω µ,h I h ω µ,h, [ξ]) Ch ω ξ + a 1 KH 1 h 3 ω H 3 [ξ] L () () { } + CH 1 ξ 1 h 3 ω H 3 + h ω [ξ] L () () Ch{ ω ξ + ω ξ } Ch r+1 ξ u r+1. From (.3), (3.1) and (3.8), we see that (ω µ ω µ,h, [ξ]) CH 1/ ω [ξ] L () CH 1/ ξ { } [ W W] L () + [W u] L () CH 1/ ξ ( η L ( ) + η ). Substituting the above two inequalities into (3.31) leads to the error estimate (3.3). Taking a time derivative of the Eq. (3.1), we can get an equation for ξ t. In analogy to the proof of (3.3), (3.4) can be proved. The proof of Lemma 3.5 is completed. Now, we turn to derive an L ( )-norm error estimate for θ n. It follows from trace theorem in [1] that ψ L () C ψ ψ 1, ψ H 1 ( ), (3.3)

12 1178 MA, SUN, AND YANG which will be used in the following proof. Lemma 3.6. For θ n defined by (3.), there holds the following error estimate max θ n 0 n N L C{ H [ ( max ( ) η n L 1 n N ( ) + ηn ) + η t L (0,T ;L ( ))] provided + ( t) + H 5 + h (r+1)}, (3.33) t C 1 H, C 1 = a 0(1 δ) C0 3, 0 <δ 1. (3.34) 16a1 K C Proof. From (.1), we get the weak formulation: ( t u n, v) + D(u n, v) + ( u n µ, [v]) = (f n + ρ n, v), v M h, (3.35) where u n (x) = u(x, t n ) and u n µ = (A un ) ν. The time truncation term ρ n satisfies N T ρ n t n ( t) u tt (, t) dt C( t). n=1 Combining (3.1) and (3.35), we have v M h, 0 ( t W n, v) + (A W n, v) + ( W n 1 µ,h, [v]) + (v µ,h, [ W n 1 ]) + a 1 KH 1 ([ W n 1 ], [v]) = ( W n 1 µ W n µ, [v]) + (v µ,h, [ W n 1 W n ]) + a 1 KH 1 ([ W n 1 W n ], [v]) + (f n + ρ n, v) ( t ξ n + cu n, v) + ( u n µ,h un µ, [v]). (3.36) Subtracting (3.36) from (.6) and taking v = θ n, we obtain ( t θ n, θ n ) + (A θ n, θ n ) + a 1 KH 1 ([θ n ], [θ n ]) + ( θ n µ,h, [θ n ] ) = ( θ n µ,h θ n 1 µ,h, [θ n ] ) + ( θ n µ,h, [θ n θ n 1 ] ) + a 1KH 1 ([θ n θ n 1 ], [θ n ]) + (A (ξ n ξ n 1 ), θ n ) + ( t ξ n + cξ n cθ n ρ n, θ n ) ( u n 1 µ,h un µ,h, [θ n ] ) + ( u n µ un µ,h, [θ n ] ). (3.37) Because and ( t θ n, θ n ) = 1 t [ θ n θ n 1 ]+ t tθ n ( θ n µ,h, [θ n θ n 1 ] ) = ( θ n µ,h, [θ n ] ) ( θ n 1 µ,h, [θ n 1 ] ) ( θ n µ,h θ n 1 µ,h, [θ n 1 ] ),

13 by summing (3.37) over n we have DDM FOR PARABOLIC EQUATION ON GENERAL DOMAIN θ n + ( t) t θ n + t [ (A θ n, θ n ) + a 1 KH 1 ([θ n ], [θ n ]) + ( θ n µ,h, [θ n ] ) ] n 1 [ t + t tθ k + (A θ k, θ k ) + a 1 KH 1 ([θ k ], [θ k ]) + ( θ k µ,h, [θ k ] ) ] = 1 θ 0 t ( θ 0 µ,h, [θ 0 ] ) + t [ (θ k µ,h θ k 1 µ,h, [θ k θ k 1 ] ) + a 1K H ([θ k θ k 1 ], [θ k ]) + (A (ξ k ξ k 1 ), θ k ) + ( t ξ k + cξ k cθ k ρ k, θ k ) ( u k 1 µ,h uk µ,h, [θ k ] ) + ( u k µ uk µ,h, [θ k ] ) ]. (3.38) Furthermore, noting that ( θ n µ,h, [θ n ] ) ε θ n 1,A + a 1K ε H 1 [θ n ] L () and taking 0 <ε= < 1, we have 1 θ n + t (1 C 0) ( ) (A θ n, θ n ) + a 1 KH 1 [θ n ] L () [ t + t ( ) ] tθ k + C 0 (A θ k, θ k ) + a 1 KH 1 [θ k ] L () 1 θ 0 + t ( θ 0 µ,h, [θ 0 ] ) + t [ (θ k µ,h θ k 1 µ,h, [θ k θ k 1 ] ) + a 1K H ([θ k θ k 1 ], [θ k ]) + (A (ξ k ξ k 1 ), θ k ) + ( t ξ k + cξ k cθ k ρ k, θ k ) ( u k 1 µ,h uk µ,h, [θ k ] ) + ( u k µ uk µ,h, [θ k ] ) ]. (3.39) We estimate the terms on the right-hand side of (3.39) one by one. From (3.3), we see that t ( θ k µ,h θ k 1 µ,h, [θ k θ k 1 ] ) t C 1/ t 1 (a 0C 0 t) 1/ [θ k θ k 1 k ] L () θ µ,h θ k 1 µ,h θ k θ k 1 1/ 1 θ k θ k 1 1/ k θ µ,h θ k 1 µ,h L () ( θ k 1 + θ k 1 1 ) θ k θ k 1 + C ( t) 3/ a0 C 0 L () a 1 K H θ k θ k 1 1,A

14 1180 MA, SUN, AND YANG ( t) 4 ( t) 4 t θ k + a 0C 0 t t θ k + C 0 t θ k 1 + 4a 1KC ( t) 3/ a0 C 0 H θ k 1,A [ θ k 1,A + a 0 θ k ] + (1 δ)c 0 t θ k 1,A (3.40) provided that t a 0(1 δ) C0 3H, 0 <δ 1. (3.41) 16a1 K C Similarly, we have a 1 K t H a 1KC 1/ t H 1 (a 0C 0 t) 1/ ( t) 4 ( t) 4 provided that ([θ k θ k 1 ], [θ k ]) a 1K t H It is easy to get [θ k θ k 1 ] L () [θ k ] L () θ k θ k 1 1/ 1 θ k θ k 1 1/ [θ k ] L () ( θ k 1 + θ k 1 1 ) θ k θ k 1 + a 1 K C ( t) 3/ a0 C 0 H t θ k + a 0C 0 t t θ k + C 0 t For the last three terms, we can get θ k 1 + a 1 K C ( t) 3/ a0 C 0 H [θ k ] L () [θ k ] L () [ θ k 1,A + a 0 θ k ] + (1 δ)c 0 t a 1K H [θ k ] L () (3.4) t a 0(1 δ) C0 3H, 0 <δ<<1. (3.43) 4a1 K C (A (ξ k ξ k 1 ), θ k ) t [ θ k 1,A + ] tξ k 1,A. (3.44) ( t ξ k + cξ k cθ k ρ k, θ k ) K{ ρ k + ξ k + t ξ k + θ k }, (3.45) ( u k µ,h uk 1 µ,h, [θ k ] ) H u k µ,h δc 0 a 1 K uk 1 µ,h L + 0.5δC () 0a 1 KH 1 [θ k ] L () C u k u k 1 1,A + 0.5δC 0a 1 KH 1 [θ k ] L (), (3.46)

15 DDM FOR PARABOLIC EQUATION ON GENERAL DOMAIN 1181 and ( u k µ uk µ,h, [θ k ] ) H u k µ δc 0 a 1 K uk µ,h L + 0.5δC () 0a 1 KH 1 [θ k ] L () CH 5 u k W, + 0.5δC ( ) 0a 1 KH 1 [θ k ] L (3.47) () by Lemma 3.. From (3.41) and (3.43), we take t = a 0(1 δ) C 3 0 H 16a 1 K C Def. = C 1 H, 0 <δ<<1. (3.48) Collecting from (3.39) to (3.48), we obtain 1 θ n + t (1 C 0) ( θ n 1,A + a ) 1KH 1 [θ n ] L () 1 θ 0 + t ( θ 0 µ,h, [θ 0 ] ) [ ( + C t t θ k 1,A + tξ k 1,A) + θ k + t ξ k + ξ k + ρ k + ( t) u k + H 5 u k W ( )]. (3.49), Noting θ 0 = 0 and (3.5), and using the discrete Gronwall Lemma and Lemma 3.4 to (3.49), we finally derive (3.33). Now, we can prove Theorem.1. Applying Lemmas 3.4 and 3.5, we have max 1 n N un U n max { ξ n + θ n } 1 n N C { h r+1 + H 1/ max 1 n N ηn L ( ) + H 1/ η t L (0,T ;L ( )) + t + H 5/}. (3.50) By (3.50), using (3.15) and (3.16), respectively, we have 1. For linear finite element space (r = 1) in two-dimensional domain (d = ) case max 1 n N un U n C{h + H 1/ h lnh +H 1/ h + t + H 5/ } provided that H = O((1 + lnh ) ).. For other finite element spaces (r > 1) case Then we can derive (.9). t C{h + H 1/ h (1 + lnh ) + t + H 5/ } C{h + t + H 5/ }, (3.51) max 1 n N un U n C{h r+1 + H 1/ h r+1 + t + H 5/ } C{h r+1 + t + H 5/ }. (3.5)

16 118 MA, SUN, AND YANG IV. CONVERGENCE ANALYSIS OF EIM-PG Because the differences between two schemes (.6) and (.8) are the second and third term to calculate the flux on the innerdomain boundary, the convergence analysis of EIM-PG scheme (.8) is similar to that of IM-PG scheme (.6). For the sake of brevity, we describe the processes of analysis for (.8) simply. The proof of the Theorem. is based on the following basic lemmas, whose proofs are similar to that of lemmas in Section III accordingly. We omit the proofs of Lemmas 4., 4.4, and 4.5 here. Lemma 4.1. For sufficiently smooth function W, there hold estimates Ŵ E W L () + 1 H W L 3 ( ) (4.1) and Ŵ E W L () CH 4 W W 4, ( ) (4.) and W(x) Ŵ E (x) = H 4 W ν 4 (x) + o(h 6 ), x on, (4.3) FIG. 3. The solution u(x, y, t) = 100tx 3 (1 x) cos(πy) at t = 0.5. [Color figure can be viewed in the online issue, which is available at

17 DDM FOR PARABOLIC EQUATION ON GENERAL DOMAIN 1183 FIG. 4. The contour of the solution u(x, y, t) = 100tx 3 (1 x) cos(πy) at t = 0.5. where W ν 4 is the fourth normal derivative of W on. Proof. By using Taylor expansions, we see that x on Ŵ E (x) W(x) = 4 H/ t W t (x + t ν )dt dt 1 H t W t (x + t ν )dt dt 3H H/ 0 6H H 0 such that Ŵ E (x) W(x) 4 H 3 [ H/ 1/ W(x + tν ) dt] + H/ + 1 H W L 3 ( ), H 3 [ H H ] 1/ W(x + tν ) dt TABLE I. L -norm error at t = 0.5 of IM-PG scheme case. Implicit IM-PG Grids h H/h e h L Rate e h L Rate e e e e e e e e e e e e

18 1184 MA, SUN, AND YANG TABLE II. L -norm error of grids on at t = 0.5 of IM-PG scheme case. IM-PG Grids h H/h Implicit L-side R-side e e e e e e e e e e e e e e e 03 which implies (4.1). Furthermore, we see that W(x) Ŵ E (x) = 4 3 (W (x) W H/(x)) 1 3 (W(x) W H (x)) which leads to (4.) and (4.3). { = ( ) H W (x) 1 ν 10 ( ) } H 4 W (x) + o(h 6 ν4 ) } { 1 6 H W ν (x) 1 10 H 4 W ν 4 (x) + o(h 6 ) = H 4 W ν 4 + o(h 6 ), (4.4) Lemma 4.. small, we have Let G ={y y = x + tν, t [ H, H ], x on }. Ifψ H 1 ( ) and H>0 is Eψ L (G) K ψ L ( ), (Eψ) ν L (G) K ψ L ( ), (4.5) where K is defined as same as (3.6). For functions ψ with restrictions in H 1 ( 1 ) H 1 ( ), we use the definition of the norm ψ = (A ψ, ψ) + a 1 KH 1 ([ψ], [ψ]). (4.6) Lemma 4.3. There exists a positive constant C 0 = 1 3 such that for small H>0, C 0 ψ (A ψ, ψ) + a 1 KH 1 ([ψ], [ψ]) + ( ψ µ,e, [ψ]), v M h. (4.7) TABLE III. The CPU time cost for the time interval [0, 0.5] of IM-PG scheme case. Grids h Implicit IM-PG e 01.04s 0.51s e s 4.4s e s 70.74s

19 TABLE IV. DDM FOR PARABOLIC EQUATION ON GENERAL DOMAIN 1185 L -norm error for each case at t = 0.5 of EIM-PG scheme case. Implicit EIM-PG Grids h H/h e h L Rate e h L Rate e e e e e e e e e e e e Proof. First, we note that ψ µ,e = 4 3H = 1 H + 1 6H H/ H/ H/ H/ ψ µ (x + tν )dt 1 6H ψ µ (x + tν )dt 1 6H ψ µ (x + tν )dt t H/ H H t H/ ψ µ (x + tν )dt ψ µ (x + ν )dt 4 [ H 1/ 3 ψ µ (x + ν ) dt]. (4.8) H H Similarly like (3.19), by (4.5) and (4.8) we can get Then, we have H ψ µ,e L () 16 9 H H ψ µ (x + tν ) dtd 16a 1K ψ 1,A 9. (4.9) (A ψ, ψ) + a 1 KH 1 ([ψ], [ψ]) + ( ψ µ,e, [ψ]) ( (1 ε)(a ψ, ψ) ) a 1 KH 1 [ψ] L 9ε. (4.10) () Taking ε = 3 in this estimate leads to (4.7). The proof of Lemma 4.3 ends. Similarly as the elliptic projection (3.1), to get optimal error estimates, we introduce an elliptic projection W M h of the solution u as follows: (A (u W), v) + ( û µ,e W µ,e, [v] ) + ( v µ,e, [u W ] ) + a 1 KH 1 ([u W ], [v]) = 0, v M h. (4.11) TABLE V. L -norm error of grids on at t = 0.5 of EIM-PG scheme case. EIM-PG Grids h H/h Implicit L-side R-side e e e e e e e e e e e e e e e 03

20 1186 MA, SUN, AND YANG TABLE VI. The CPU time cost for the time interval [0, 0.5] of EIM-PG scheme case. Grids h Implicit EIM-PG e 01.04s 1.1s e s 4.64s e s 8.34s It follows from Lemma 4.3 that the projection problem (4.11) has unique solution for small H. Let The following lemma gives the bounds of ξ n. ξ n = u n W n, θ n = U n W n. (4.1) Lemma 4.4. There hold a priori estimates: ξ L ( ) C{h r+1 + H 1/ η L ( )} (4.13) and ξ t L ( ) C{h r+1 + H 1/ η t L ( )}. (4.14) Now, we turn to derive an L ( )-norm error estimate for θ n. Lemma 4.5. For θ n defined by (4.1), there exists the following error estimate max 1 n N θ n L ( ) C { H ( max 1 n N ηn L ( ) + η t L (0,T ;L ( )) ) + h (r+1) + ( t) + H 9}, (4.15) FIG. 5. The domain with a skew line interdomain boundary.

21 DDM FOR PARABOLIC EQUATION ON GENERAL DOMAIN 1187 FIG. 6. The solution u(x, y, t) = (t + 1)(x x 4 ) (y y ) at t = 0.5. [color figure can be viewed in the online issue, which is available at provided Here, the main equation for θ is t C 1 H, C 1 = a 0(1 δ) C 0 3, 0 <δ<<1. (4.16) 16a1 K C ( t θ n, v) + (A θ n, v) + ( θ n µ,e, [v]) + ( v µ,e, [θ n ]) + a 1 KH 1 ([θ n ], [v]) = ( θ n µ,e θ n 1 µ,e, [v]) + ( v µ,e, [θ n θ n 1 ]) + a 1 KH 1 ([θ n θ n 1 ], [v]) + (A (ξ n ξ n 1 ), v) + ( t ξ n + cξ n cθ n ρ n, v) ( û n 1 µ,e ûn µ,e, [v]) + ( u n µ ûn µ,e, [v]). (4.17) TABLE VII. L -norm error at t = 0.5 of IM-PG skew line scheme case. Implicit IM-PG Grids h H/h e h L Rate e h L Rate e e e e e e e e e e e e

22 1188 MA, SUN, AND YANG TABLE VIII. L -norm error of grids on at t = 0.5 of IM-PG skew line scheme case. IM-PG Grids h H/h Implicit L-side R-side e e e e e e e e e e e e e e e 04 Similarly, as the proofs of (3.50) and (3.51), applying Lemmas 4.4, 4.5, and the condition for H, we can derive (.11). The proof of Theorem. is completed. Remark. From Theorem., we can know that the EIM-PG scheme has an accuracy of higher order for H than that of IM-PG scheme, and has the same accuracy order as that of the implicit method with respect to t and h. This shows that EIM-PG scheme can use larger width H of middle strip domain than that of IM-PG scheme so that the time-step constraint is more weaker than that of IM-PG scheme. V. NUMERICAL EXPERIMENTS In this section, we present some numerical experiments for the procedures described above. Example 1. IM-PG scheme case. Let =[0, 1] [0, 1]. The problem considered here is just the second model in [6]: { ut u = f(x, y, t), in (0, T ], u 0 (5.1) (x, y) = 0, in, t = 0, where f(x, y, t) is chosen so that u(x, y, t) = 100tx 3 (1 x) cos(πy) (see Figs. 3 and 4). We consider two scenarios: (1) fully implicit Galerkin method on uniform mesh; i.e. no domain decomposition; () Galerkin domain decomposition method on global uniform mesh with two equal subdomains 1 = (0, 1 ) (0, 1), = ( 1,1) (0, 1), with the interdomain boundary ={ 1 } (0, 1). In these runs, the solution u is approximated in the space of continuous piecewise bilinear function. We approximate (5.1) by using 4-node quadrilateral mesh on 0 0, and grids, respectively. For each domain decomposition case, we take H 5/ = h to balance error accuracy with respect to h and H and mesh ration t =.5H 5/. In Table I, we give the L -norm error for e h = u U at time t = 0.5. Hereafter, Implicit means the fully implicit Galerkin approximate solution and L(R)-side stands for the left(right)-hand side of. TABLE IX. The CPU time cost for the time interval [0, 0.5] of IM-PG skew line scheme case. Grids h Implicit IM-PG e s 0.47s e s 3.5s e s 67.1s

23 TABLE X. DDM FOR PARABOLIC EQUATION ON GENERAL DOMAIN 1189 L -norm error at t = 0.5 of biquadratic element IM-PG case. Implicit IM-PG Grids h H/h e h L Rate e h L Rate e e e e e e e e e e e e And we present the L -norm error of grids on the interface at t = 0.5 (see Table II). Remark 3. From Tables I and II, we can know that the IM-PG scheme approximates the exact solution better than fully implicit Galerkin method, having second-order convergence in h. Furthermore, time-step constraint of IM-PG scheme is less severe than that which comes with a fully explicit method. Next, we compare the CPU time cost of the fully implicit Galerkin method and IM-PG scheme for the time interval [0, 0.5] (see Table III). Remark 4. From Table III, we can see that the CPU time cost of IM-PG scheme is smaller than that of implicit scheme for each case, and when the mesh is finer, because the system becomes larger, IM-PG scheme exhibits its superiority. Example. EIM-PG scheme case. We approximate (5.1) on 0 0, 40 40, and grids by using EIM-PG scheme. For each domain decomposition case, we take H 9/ = h to balance error accuracy with respect to h and H. The mesh ration is t = 5H 9/. Table IV shows the L -norm error for e h = u U at time t = 0.5. And Table V lists the L -norm error of grids on the interface at t = 0.5. Table VI shows the comparison the CPU time cost for the time interval [0, 0.5]. Remark 5. From Tables IV and V, we can see that the EIM-PG scheme also approximates the exact solution better than fully implicit Galerkin method, having second-order convergence in h. It is shown that EIM-PG scheme can use larger middle strip domain s width H than that of IM-PG scheme. And they have the same time-step constraint, which is still less severe than that which comes with a fully explicit method. From Table VI, we also can see that the CPU time cost of EIM-PG is smaller than that of implicit scheme for each case and a little more than that of IM-PG scheme. This shows that the costs of the extrapolation procedures are little. When the mesh is finer, because the system becomes larger, EIM-PG scheme exhibits its superiority. TABLE XI. L -norm error of grids on at t = 0.5 of biquadratic element IM-PG case. IM-PG Grids h H/h Implicit L-side R-side e e e e e e e e e e e e e e e 04

24 1190 MA, SUN, AND YANG TABLE XII. The CPU time cost for the time interval [0, 0.5] of biquadratic element IM-PG case. Grids h Implicit IM-PG e s 0.4s e s 8.5s e s s Example 3. IM-PG scheme for a skew line interdomain boundary case. We consider the following problem on =[0, 4] [0, ], with a skew line interdomain boundary ={(x, y) y = x 1}: { ut u = f(x, y, t), in (0, T ], ( u 0 (x, y) = x x 4 ) ( y y ), in, t = 0, (5.) ( ) ( ) where f(x, y, t) is chosen so that the solution u(x, y, t) = (t + 1) x x y y 4 (see Figs. 5 and 6). We approximate (5.) with uniform three-node rect-triangle elements on 40 0, 80 40, and grids by IM-PG scheme. Similarly, like Example 1, we take H 5/ = h to balance error accuracy with respect to h and H and mesh ration t =.5H 5/. The L -norm errors for e h = u U at time t = 0.5 are given in Table VII below. Table VIII shows the L -norm error of grids on the interface at t = 0.5. FIG. 7. The domain with four subdomains.

25 DDM FOR PARABOLIC EQUATION ON GENERAL DOMAIN 1191 FIG. 8. The solution u(x, y, t) = e t cos(πx)cos(πy) at t = 0.5. [color figure can be viewed in the online issue, which is available at Table IX shows the comparison the CPU time cost for the time interval [0, 0.5]. Remark 6. From Tables VII XI, we can still get the same concludes as Remarks 3 and 4, proving that IM-PG scheme can be used for a skew line interdomain boundary case. Example 4. IM-PG scheme with biquadratic finite element case. To show the methods of this article valid for higher order finite elements, for simplicity, we consider the IM-PG scheme with biquadratic finite element for the problem (5.1). Here, problem (5.1) is approximates by normal nine-node quadrilateral elements on 10 10, 0 0, and grids, respectively. In these runs, we still take H 5/ = h 3 to balance error accuracy with respect to h and H and mesh ration t =.5H 5/. In Table X, we give the L -norm error for e h = u U at time t = 0.5. And we present the L -norm error of grids on the interface at t = 0.5 in Table XI. Next, Table XII shows the CPU time cost for the time interval [0, 0.5]. TABLE XIII. L -norm error at t = 0.5 of four subdomains IM-PG case. Implicit IM-PG Grids h H/h e h L Rate e h L Rate e e e e e e e e e e e e

26 119 MA, SUN, AND YANG TABLE XIV. L -norm error of grids on at t = 0.5 of four subdomains IM-PG case. IM-PG Grids h H /h e e e e e e e e e e e e e e e e e e 03 Remark 7. From Tables X XII, we can get the same concludes as that of Tables I III, proving that IM-PG scheme with biquadratic finite element works well. Example 5. IM-PG scheme case for four subdomains. Let =[0, 1] [0, 1] be divided equally into four subdomains by i, i = 1,..., 4 with four interdomain boundarys i, i = 1,..., 4 (see Fig. 7). The problem considered here is as follows: { ut u + u = f(x, y, t), in (0, T ], u 0 (5.3) (x, y) = 0, in, t = 0, where f(x, y, t) is chosen so that u(x, y, t) = e t cos(πx)cos(πy) (see Fig. 8). Problem (5.3) is approximated by using 4-node quadrilateral elements on 0 0, and grids, respectively. For each domain decomposition case, we still take H 5/ = h to balance error accuracy with respect to h and H. The mesh ration is t =.5H 5/. Table XIII gives the L -norm error for e h = u U at time t = 0.5. Table XIV shows the maximus absolute error between approximate solutions of two sides for grids on the interface i, i = 1,...,4att = 0.5. Table XV compares the CPU time cost for the time interval [0, 0.5]. Remark 8. From Tables XIII and XV, we can get the same concludes as that of Tables I III, proving that IM-PG scheme works well for four sub-domians. VI. CONCLUSION AND PERSPECTIVE We have presented parallel Galerkin domain decomposition procedures for parabolic equation on general domain, which use implicit Galerkin method in the subdomains and simple explicit flux calculation on the interdomain boundaries by integral mean method or extrapolation method to predict the inner-boundary conditions. L -norm error estimates are derived for these procedures. Compared with [6], these L -norm error estimates avoid the loss of H 1/ factor. Experimental results are presented to confirm theoretical results. TABLE XV. The CPU time cost for the time interval [0, 0.5] of four subdomains IM-PG case. Grids h Implicit IM-PG e 01.83s 0.68s e s 4.87s e s 95.65s

27 DDM FOR PARABOLIC EQUATION ON GENERAL DOMAIN 1193 The importance of this article is the use of integral mean methods on general domain to avoid the loss of H 1/ factor in L -norm error estimates. For the sake of simplicity, we consider the homogeneous Neumann boundary condition here. The methods used in this article can be easily extended to Dirichlet or nonhomogenous Neumman boundary condition by adding some integral terms on the boundary and similar results can be derived. Furthermore, the theoretical results of this article are still valid for using different Galerkin finite element spaces on subdomains, i.e., discontinuous Galerkin method on the interdomain boundaries. We will present the results for these cases in another article. The authors thank the referees for their valuable suggestions and constructive comments. References 1. J. H. Bramble, J. E. Pasciak, and J. Xu, Parallel multilevel preconditioners, Math Comp 55 (1990), 1.. J. H. Bramble, J. E. Pasciak, and J. Xu, Convergence estimates for product iterative methods with application to domain decomposition, Math Comp 57 (1991), X. C. Cai, Additive Schwarz algorithms for parabolic convection-diffusion equations, Numer Math 60 (1991), X. C. Cai, Multiplicative Schwarz methods for parabolic problem, SIAM J Sci Comput 15 (1994), X. C. Cai, Some domain decomposition algorithms for nonselfadjont elliptic and parabolic partial differential equations, Ph.D. Thesis, Courant Institute, New York, C. N. Dawson and T. F. Dupont, Explicit/implicit conservative Galerkin domain decomposition procedures for parabolic problems, Math Comp 58 (199), M. Dryja and O. B. Widlund, An additive variant of Schwarz alternating methods for many subregions, Technical Report 339, Department of Computer Science, Courant Institute, New York, J. Xu, Theory of multilevel methods, Ph.D. Thesis, Cornell University, Ithaca, NY, J. Xu, Iterative methods by space decomposition and subspace correction: a unifying approach, SIAM Rev 34 (199), H. Rui and D. P. Yang, Schwarz type domain decomposition algorithms for parabolic equations and error estimates, Acta Math Appl Sin 14 (1998), H. Rui and D. P. Yang, Multiplicative Schwarz algorithm with time stepping along characteristics for convection diffusion equations, J Comp Math 19 (001), P. L. Lions, On Schwarz alternating method, Part I, R. Glowinski, G.H. Golub, G.A. Meurant, and J. Périaux, (editors), Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1988, pp P. L. Lions, On Schwarz alternating method, Part II, T. Chan, R. Glowinski, J. Périaux, and O.B. Widlund, editors, Proceedings of the Second International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1989, pp T. Lu, T. M. Shih, and C. B. Liem, Parallel algorithms for variational inequalities based on domain decomposition, J Comp Math 9 (1991), X. C. Tai, A space decomposition method for parabolic equations, Numer Methods Partial Differential Eq 14 (1998), J. H. Bramble and A. H. Schatz, Higher order local accuracy by averaging in the finite element method, Math Comp 31 (1977),

28 1194 MA, SUN, AND YANG 17. P. G. Ciarlet, The finite element method for elliptic problems, North-Holland Publ., Amsterdam, J. Nitsche, L -error analysis for finite elements, J.R. Whiteman, editor, Mathematics of finite elements and applications III, Academic Press, New York, 1979, pp A. H. Schatz and L. B. Wahlbin, Maximun norm estimates in the finite element method on plane polygonal domains, I Math Comp 3 (1978), A. H. Schatz and L. B. Wahlbin, Maximun norm estimates in the finite element method on plane polygonal domains, II, Math Comp 33 (1979), S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods (Texts in applied mathematics 15), Springer-Verlag, New York, 1996.