Maximum norm error analysis of a nonmatching ... - Semantic Scholar
Applied Mathematics and Computation 238 (2014) 21–29
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Maximum norm error analysis of a nonmatching grids finite element method for linear elliptic PDEs q Messaoud Boulbrachene a,⇑, Qais Al Farei b a b
Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Muscat 123, Oman Higher College of Technology, IT Department, Muscat 123, Oman
a r t i c l e
i n f o
a b s t r a c t In this paper, we study a nonmatching grid finite element approximation of linear elliptic PDEs in the context of the Schwarz alternating domain decomposition.We show that the approximation converges optimally in the maximum norm, on each subdomain, making use of the geometrical convergence of both the continuous and corresponding discrete Schwarz sequences. We also give some numerical results to support the theory. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Elliptic PDEs Schwarz alternating method Nonmatching grids Finite element L1 – error estimate
1. Introduction The Schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. Extensive analysis of Schwarz alternating method for linear elliptic boundary value problems can be found in [1,2].The effectiveness of Schwarz methods for these problems, especially those in fluid mechanics, has been demonstrated in many papers. For that, we refer to the proceedings of the annual domain decomposition conference beginning with [3]. In this paper, we are interested in the error analysis in the maximum norm of a finite element Schwarz alternating method for linear elliptic problems on two overlapping subdomains with nonmatching grids: we consider a domain X which is the union of two overlapping subdomains where each subdomain has its own triangulation. This kind of discretization is very interesting as they can be applied to solving many practical problems which cannot be handled by global discretizations.They are earning particular attention of computational experts and engineers as they allow the choice of different mesh sizes and different orders of approximate polynomials in different subdomains according to the different properties of the solution and different requirements of the practical problems. For other kinds of discretization of elliptic PDEs, we refer to [4–6]. More specifically, we show that the approximation converges optimally on each subdomain, developing a new approach which consists of combining the geometrical convergence of both the continuous and discrete Schwarz sequences with an estimate in the maximum norm between the continuous and discrete Schwarz iterates. More precisely, if ðuni; Þ denotes the Schwarz sequence in the subdomain Xi and ðuni;hi Þ is its finite element counterpart with respect to the triangulation with meshsize hi , we show that
n ui uni;hi
L1 ðXi Þ
q
2
6 Ch jlog hj
The authors would like to thank the referee for his careful reading and valuable comments.
⇑ Corresponding author.
E-mail addresses: [email protected] (M. Boulbrachene), [email protected] (Q. Al Farei). http://dx.doi.org/10.1016/j.amc.2014.03.146 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
ð1:1Þ
22
M. Boulbrachene, Q. Al Farei / Applied Mathematics and Computation 238 (2014) 21–29
and
2 ui ui;h 1 6 Ch jlog hj i L ðX Þ i
ð1:2Þ
where ui denotes the true solution on Xi ; ui;hi is the uniform limit of the discrete Schwarz sequence ðuni;hi Þ, and C is a constant independent of n and h. For other works on maximum norm analysis of overlapping nonmatching grid methods for elliptic problems (cf. e.g., [8,9]). Now, we give an outline of the paper. In Section 2, we state the continuous Schwarz sequences and prove their geometrical convergence. In Section 3, we define their respective finite element counterparts in the context of nonmatching overlapping grids. Section 4 discusses the L1 - error analysis of the method, and Section 5 is devoted to some numerical experiments. 2. The continuous problem Consider the second order linear elliptic problem:
Du þ cu ¼ f
in X
u¼0
on @ X
ð2:1Þ
where X is a bounded polyhedral domain of R2 (or R3 Þ with boundary @ X; c a nonnegative function and f a given regular function. We decompose the domain X into two overlapping polyhedral subdomains X1 and X2 such that:
X ¼ X1 [ X2 and
X1 \ X2 – ; Let ci ¼ @ Xi \ Xj ; i; j ¼ f1; 2g; i – j denote the internal boundaries of Xi , and Ci ¼ @ Xi n ci the external boundaries such that i \ c j ¼ ;. Let also fi ¼ f =X . We associate with this problem the following Schwarz problem: find @ Xi ¼ Ci [ ci , and c i fu1 ; u2 g such that
8 > < Dui þ cui ¼ fi ui ¼ 0 > : ui ¼ uj
in Xi on Ci
ð2:2Þ
on ci
and
ui ¼ u=Xi Next, we define the associated Schwarz sequences and prove that they converge geometrically to the unique solution fu1 ; u2 g of system (2.2). 2.1. The continuous Schwarz sequences nþ1 Let u0 be an initial guess in CðXÞ where u02 ¼ u0 =X2 . We define the continuous Schwarz sequences ðunþ1 1 Þ in X1 and ðu2 Þ in X2 , by solving the following two subproblems:
8 nþ1 nþ1 > < Du1 þ cu1 ¼ f1 nþ1 u1 ¼ 0 > : nþ1 u1 ¼ un2
in X1 on C1 on c1
ð2:3Þ
and
8 nþ1 nþ1 > < Du2 þ cu2 ¼ f2 nþ1 u2 ¼ 0 > : nþ1 u2 ¼ unþ1 1
in X2 on C2
ð2:4Þ
on c2
respectively. Next, we shall prove that these Schwarz sequences converge geometrically to the solution of problem (2.2). For that let us define the following fixed point mapping. 2.1.1. A fixed point mapping associated with the Schwarz Method Indeed, given wj 2 CðXj Þ, let us first consider the following Dirichlet problem
M. Boulbrachene, Q. Al Farei / Applied Mathematics and Computation 238 (2014) 21–29
8 i þ cu i ¼ 0 in Xi > < Du on Ci ui ¼ 0 > : on ci ui ¼ wj
23
ð2:5Þ
For i; j 2 f1; 2g; i – j, let Ei ¼ Cðci Þ, we define the mappings:
T i : Ej # Ei
ð2:6Þ
v j # T i ðv j Þ where
i =c T i ðv j Þ ¼ u j
ð2:7Þ
Lemma 1 (see [8]). The mappings T i ; j ¼ 1; 2, are contractions, that is,
kT i kLðL1 ðcj Þ;L1 ðci ÞÞ ¼ qi < 1 Notation 1. From now and on, we shall adopt the following notations:
kkL1 ðX1 Þ ¼ kk1 ; kkL1 ðc1 Þ ¼ jj1 ;
kkL1 ðX2 Þ ¼ kk2 kkL1 ðc2 Þ ¼ jj2
pi ðv Þ ¼ v =ci 8v 2 CðXi Þ nþ1 Proposition 1. The sequences ðunþ1 1 Þ and ðu2 Þ; n P 0 produced by the Schwarz alternating method converge geometrically, in the maximum norm, to the solutions fu1 ; u2 g of problem (2.2) More precisely, we have
nþ1 u u1 1 6 qn1 qn2 u02 u2 2 1
ð2:8Þ
nþ1 0 u u u2 u2 2 6 qn1 qnþ1 2 2 2 2
ð2:9Þ
and
where
q ¼ max fq1 ; q2 g Proof. We prove this lemma by induction. Let us first prove (2.8). Indeed, subtracting (2.2) from (2.3) and (2.4), respectively, we get
8 nþ1 nþ1 > < Dðu1 u1 Þ þ cðu1 u1 Þ ¼ 0 in X1 unþ1 u1 ¼ 0 on C1 1 > : nþ1 n u1 u1 ¼ u2 u2 on c1 and
8 nþ1 nþ1 > < Dðu2 u2 Þ þ cðu2 u2 Þ ¼ 0 in X2 unþ1 u2 ¼ 0 on C2 2 > : nþ1 nþ1 u2 u2 ¼ u1 u1 on c2 n ¼ 1: Using both the definition of both T 1 and T 2 and the maximum principle, we have
2 u u1 6 u1 u2 6 T 1 ðp2 ðu1 u1 ÞÞ 6 q u1 u1 6 q T 2 ðp1 ðu0 u2 ÞÞ 6 q q u0 u2 1 1 1 1 2 2 1 2 1 2 1 1 1 2 2 1 n ¼ 2:
3 u u1 6 u2 u2 6 T 1 ðp2 ðu2 u1 ÞÞ 6 q1 u2 u1 6 q1 T 2 ðp1 ðu1 u2 ÞÞ 6 q1 q2 u1 u2 1 2 1 1 2 2 1 1 1 2 2 1 6 q1 q2 T 1 ðp2 ðu11 u1 ÞÞ1 6 q21 q2 u11 u1 2 6 q21 q2 T 2 ðp1 ðu02 u2 ÞÞ2 6 q21 q22 u02 u2 1
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M. Boulbrachene, Q. Al Farei / Applied Mathematics and Computation 238 (2014) 21–29
So, by induction, we get
nþ1 u u1 1 6 qn1 qn2 u02 u2 1 1 Now, we can prove (2.9) in a similar way. Indeed, we have
1 u u2 6 u1 u1 6 T 2 ðp1 ðu0 u2 ÞÞÞ 6 q u0 u2 2 2 2 1 2 2 2 2 1 and
2 u u2 6 u2 u1 6 T 2 ðp1 ðu1 u2 ÞÞ 6 q2 u1 u2 6 q2 T 1 ðp2 ðu1 u1 ÞÞ 6 q1 q2 u1 u1 2 1 2 2 1 1 2 2 2 1 1 2 6 q1 q2 T 2 ðp1 ðu02 u2 ÞÞ2 6 q1 q22 u02 u2 1 Hence, we get
nþ1 0 u u u2 u2 2 6 qn1 qnþ1 2 2 2 1 Now, taking q ¼ maxðq1 ; q2 Þ, we get
nþ1 u u1 1 6 q2n u02 u2 2 1
and
nþ1 u u2 2 6 q2nþ1 u02 u2 2 2
3. Nonmatching grids discretization For i ¼ 1; 2, let shi be a standard quasi-uniform regular finite element triangulation in Xi ; hi being its mesh size. In order to construct an overlapping nonmatching grids discretization, we assume that the two triangulation are mutually independent on X1 \ X2 in the sense that a triangle belonging to one triangulation does not necessarily belong to the other one. To define the discrete Schwarz problems on overlapping nonmatching discretization, we introduce the finite element spaces
i Þ \ H1 ðXi Þ : v h j 2 P1 8K 2 shi g V hi ¼ f v h 2 C 0 ð X K 0 where V hi is the space of continuous piecewise linear functions, and P 1 is the space of polynomials with degree less than or equal to 1. Let pi be trace operator on ci and Ehi denote the range of pi defined on V hj . We also introduce the bilinear form:
Z
ai ðu; v Þ ¼
ðru:rv þ cuv Þ:dx
Xi
and the linear form
ðu; v Þ ¼
Z
u:v :dx
Xi
We, finally, denote by r hi , the usual interpolation operator on Ehi . The discrete maximum principle assumption: we assume that the matrices whose coefficients are ai ðul ; us Þ are Matrices [10,11]. (a matrix A is said to be an Matrix if its inverse A1 exists and satisfies A1 P 0). 3.1. The discrete problem We define the discrete analog of system (2.2) as follows: find fu1;h1 ; u2;h2 g such that
8 < ai ðui;hi ; v h Þ ¼ ðfi ; v h Þ 8 : ui;h ¼ uj;h i j
vh 2 Vh
i
ð3:1Þ
on ci
3.2. The discrete Schwarz sequences The corresponding discrete Schwarz problems are given by: find unþ1 1;h1 such that
8 8 < a1 ðunþ1 1;h1 ; v h Þ ¼ ðf1 ; v h Þ : unþ1 ¼ r ðp ðun ÞÞ 1 h1 2;h2 1;h1
vh 2 Vh
on c1
1
ð3:2Þ
25
M. Boulbrachene, Q. Al Farei / Applied Mathematics and Computation 238 (2014) 21–29
and find unþ1 such that 2;h2
(
a2 ðunþ1 8 2;h2 ; v h Þ ¼ ðf2 ; v h Þ nþ1 unþ1 2;h2 ¼ r h2 ðp2 ðu1;h1 ÞÞ
vh 2 Vh
2
ð3:3Þ
on c2
As in the continuous case, we are able to prove that the above Schwarz sequences converge geometrically, in the maxi mum norm, to the solutions u1;h1 ; u2;h2 of (3.1). For that, let us define the discrete analogue of problem (2.5): Given i;hi such that wj 2 CðXj Þ, find u
i;hi ; v h Þ ¼ 0 ai ðu 8 v h 2 V hi i;hi ¼ r hi ðpi ðwj ÞÞ on ci u
and the discrete fixed point mapping T hi
j Þ # Ehi T hi : Cðc
ð3:4Þ
v j # T h ðv j Þ i
where
T hi ð v i Þ ¼ r hi
pj ðui;hi Þ
ð3:5Þ
Lemma 2 [8]. Let the discrete maximum principle hold. Then, there exists a constant 0 < li < 1 independent of h such that
T h 1 i LðL ðc
jÞ ;L
1
ðci ÞÞ
6 li
nþ1 Proposition 2. Let Lemma 2 hold. Then the discrete sequences ðunþ1 1;h1 Þ; ðu2;h2 Þ converge geometrically, in the maximum norm, to the discrete solutions u1;h1 ; u2;h2 of (3.1), that is,
nþ1 n n u1;h1 u1;h1 6 l1 :l2 u02;h2 u2;h2 1
ð3:6Þ
2
and
nþ1 n nþ1 u2;h2 u2;h2 6 l1 l2 u02;h2 u2;h2 2
2
ð3:7Þ
Proof. The proof is similar to that of Proposition 1. Indeed, it suffices to use the definition of T hj and the discrete maximum principle. h
4. Maximum norm error Next, we shall estimate the error between the continuous Schwarz sequence and its discrete analogue. For that, we need to introduce the following auxiliary problems
(
~ nþ1 a1 ðu 8 1;h1 ; v h Þ ¼ ðf1 ; v h Þ
vh 2 Vh
1
vh 2 Vh
2
n ~ nþ1 u 1;h1 ¼ r h1 ðp1 ðu2 ÞÞ
ð4:1Þ
and
(
~ nþ1 a2 ðu 8 2;h2 ; v h Þ ¼ ðf2 ; v h Þ nþ1 ~ nþ1 u 2;h2 ¼ r h2 ðp2 ðu1 ÞÞ
ð4:2Þ
where un1 and un2 are the continuous Schwarz sequences. Lemma 3. Let h ¼ maxfh1 ; h2 g. Then we have nþ1 1l nþ1 2 Ch jlog hj u1 unþ1 1;h1 6 1 1l
ð4:3Þ
nþ2 1l nþ1 2 Ch jlog hj u2 unþ1 2;h2 6 2 1l
ð4:4Þ
and
where C is a constant independent of h and n.
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M. Boulbrachene, Q. Al Farei / Applied Mathematics and Computation 238 (2014) 21–29
Proof. The proof will combine the discrete maximum principle and standard finite element maximum norm error estimate for linear PDEs (see [7]). Indeed, by subtracting problems (3.2) from (4.1) and (3.3) from (4.2), we respectively get
(
nþ1 ~ nþ1 a1 ðu 1;h1 u1;h1 ; v h Þ ¼ 0
~ nþ1 u 1;h1
unþ1 1;h1
¼ r h1 ð p
n 1 ðu2
un2;h2 ÞÞ
un1;h1 ÞÞ
8
vh 2 Vh
8
vh 2 Vh
1
and
(
nþ1 ~ nþ1 a1 ðu 2;h2 u2;h2 ; v h Þ ¼ 0
~ nþ1 u 2;h2
unþ1 2;h2
¼ r h2 ð p
n 2 ðu1
2
So, by induction, we have for [n ¼ 0: domain 1]
1 ~1 0 2 2 1 0 0 0 ~ 11;h u1 u11;h1 6 u11 u þ u 1;h1 u1;h1 6 Ch jlog hj þ r h1 ðp1 ðu2 u2;h2 ÞÞ 6 Ch jlog hj þ u2 u2;h2 1 1
1
1
2
2
1
2
2
6 Ch jlog hj þ Ch jlog hj 6 Ch jlog hj [n ¼ 0: domain 2]
1 ~1 2 1 1 1 ~ 12;h þ u 6 Ch log h þ r ð p ðu u ÞÞ u j j u2 u12;h2 6 u12 u 2 h 2;h 2;h 1 1;h 2 2 2 2 1 2 2 2 2 2 2 2 0 0 0 6 Ch jlog hj þ T h2 ðr h1 ðp1 ðu2 u2;h2 ÞÞ 6 Ch jlog hj þ lrh1 ðp1 ðu2 u02;h2 ÞÞ 6 Ch jlog hj þ lu02 u02;h2 2
2
2
1
2
2
6 Ch jlog hj þ l:Ch jlog hj 6 ð1 þ lÞCh jlog hj [n ¼ 1: domain 1]
2 ~2 2 2 2 1 1 1 1 ~ 21;h u1 u21;h1 6 u21 u þ u 1;h1 u1;h1 6 Ch jlog hj þ r h1 ðp1 ðu2 u2;h2 ÞÞ 6 Ch jlog hj þ p1 ðu2 u2;h2 Þ 1 1 1 1 1 1 2 2 2 6 Ch jlog hj þ T h1 ðr h2 ðp2 ðu11 u11;h1 ÞÞ 6 Ch jlog hj þ lrh2 ðp2 ðu11 u11;h1 Þ 6 Ch jlog hj þ lu11 u11;h1 1
2
2
2
1
2
6 Ch jlog hj þ lCh jlog hj 6 ð1 þ lÞCh jlog hj [n ¼ 1: domain 2]
2 ~2 2 2 2 2 2 1 1 ~ 22;h u2 u22;h2 6 u22 u þ u 2;h2 u2;h2 6 Ch jlog hj þ r h1 ðu1 u1;h1 Þ=c2 6 Ch jlog hj þ T h2 ðp2 ðu2 u2;h2 Þ 2 2 2 2 2 2 2 2 2 2 6 Ch jlog hj þ lph2 ðu12 u12;h2 Þ=c1 6 Ch jlog hj þ lu12 u12;h2 6 Ch jlog hj þ l½ð1 þ lÞCh jlog hj 1
2
2
2
6 ð1 þ l þ l ÞCh jlog hj [n ¼ 2; domain 1]
3 ~3 2 3 2 2 ~ 31;h þ u 6 Ch log h þ r ð p ðu u ÞÞ u j j u1 u31;h1 6 u31 u 1 h 1;h 1;h 2 2;h 1 1 1 1 2 1 1 1 1 2 2 2 2 2 2 6 Ch jlog hj þ T h1 ðr h2 ðp2 ðu1 u1;h1 ÞÞÞ 6 Ch jlog hj þ lr h2 ðp2 ðu1 u21;h1 ÞÞ 6 Ch jlog hj þ lu21 u21;h1 1
2
2
2
2
1
2
6 Ch jlog hj þ l½ð1 þ lÞCh jlog hj 6 ð1 þ l þ l ÞCh jlog hj [n ¼ 2: domain 2]
3 ~3 2 3 3 3 ~ 32;h u2 u32;h2 6 u32 u þ u 2;h2 u2;h2 6 Ch jlog hj þ r h2 ðp2 ðu1 u1;h1 ÞÞ 2 2 2 2 2 2 2 2 6 Ch jlog hj þ T h2 ðr h1 ðp1 ðu22 u22;h2 ÞÞÞ 6 Ch jlog hj þ lr h1 ðp1 ðu22 u22;h2 ÞÞ 6 Ch jlog hj þ lu22 u22;h2 2
2
2
2
1
2
3
2
6 Ch jlog hj þ l½ð1 þ l þ l ÞCh jlog hj 6 ð1 þ l þ l þ l ÞCh jlog hj So, by induction nþ1 1l nþ1 2 n 2 2 6 ð1 þ l þ l þ þ l ÞCh log h 6 Ch jlog hj j j u1 unþ1 1;h1 1 1l
and nþ2 1l nþ1 2 nþ1 2 2 ÞCh jlog hj 6 Ch jlog hj u2 unþ1 2;h2 6 ð1 þ l þ l þ þ l 2 1l
which completes the proof.
h
2
27
M. Boulbrachene, Q. Al Farei / Applied Mathematics and Computation 238 (2014) 21–29
Theorem 1. There exists a constant C independent of both n and h such that:
u1 u1;h 6 Ch2 jlog hj 1 1
ð4:5Þ
u2 u2;h 6 Ch2 jlog hj 2 1
ð4:6Þ
and
Proof. Let q ¼ max fq1 ; q2 g and l ¼ max fl1 ; l2 g. Then, using Propositions 1 and 2, and Lemma 3, we have nþ1 1l nþ1 2 2n 2n 0 u1 u1;h 6 u1 unþ1 þ unþ1 u2 u2 2 þ Ch jlog hj þ l u02;h u2;h unþ1 1 1 1;h þ u1;h u1;h 6 q 1 1 1 1 1l 2
So, by taking the limit, as n ! 1, we obtain
u1 u1;h 6 1
1 2 h jlog hj 1l
which yields (4.5). The proof of (4.6) is similar and will thus be omitted.
h
Remark 1. For computational purposes, it is interesting to estimate the error between the true solution and the discrete Schwarz sequence. Corollary 1. We have
2 ui uni;h 6 qn u02 u2 2 þ Ch jlog hj 1
Proof. Indeed,
ui uni;h 6 ui uni 2 þ uni unih 2 1
2 qn u02 u2 2 þ Ch jlog hj
5. Numerical experiments Our aim in this section is to validate the theoretical results of this paper. For that, let us consider the following elliptic problem:
Du ¼ f
in X ¼ ð0; 2Þ ð0; 2Þ
u¼0
on @ X
ð5:1Þ
where f ðx; yÞ ¼ 4ðx þ yÞ 2ðx2 þ y2 Þ is the forcing term and the exact solution is uExact ¼ xyðx 2Þðy 2Þ. We decompose the domain X into two overlapping rectangular subdomains X1 and X2 , supporting independent piecewise finite element triangulation; h1 is the mesh size of the triangulation in X1 , and h2 is the mesh size of the triangulation in X2 . cWe denote by Iter the number of Schwarz iterations on the subdomain Xi , by ERRhi the error between the exact solution and the Schwarz iterate on the subdomain Xi , i.e.,
Iter ERRhi ¼ uExact uIter i;hi ¼ uExact ui;hi i
L1 ðXi Þ
and by pi is the convergence order on the subdomain Xi , determined by the formula
pi ¼ 2
lnðERRhi Þ lnðERRh0i Þ ln ðNDFÞhi ln ðNDFÞh0
ð5:2Þ
i
where ðNDFÞhi refers to the number of degrees of freedom related to hi . The computation is carried out with a tolerance e ¼ 105 , that is
ERRhi 6 e:
28
M. Boulbrachene, Q. Al Farei / Applied Mathematics and Computation 238 (2014) 21–29 Table 1 Numerical results for two subdomains with overlap equal to 3/4.
h2
1 16 1 32
1 32 1 64
1 64 1 128
Iter ERRh1 ERRh2 p1 p2 NDF h1 NDF h2
7 0:000993274 0:00138449 2:11433 2:04483 627 3445
7 0:000248486 0:000346446 2:06139 2:02376 2405 13545
7 6:21738E005 8:66422E005 2:03001 2:01205 9417 53713
h1
Table 2 Numerical results for two subdomains with overlap equal to 1/2.
h2
1 16 1 32
1 32 1 64
1 64 1 128
Iter ERRh1 ERRh2 p1 p2 NDF h1 NDF h2
9 0:00114442 0:00194233 2:12042 2:04926 627 2925
9 0:000287017 0:000486318 2:05766 2:02542 2405 11481
9 7:21256E005 0:000121875 2:02369 2:0103 9417 45489
h1
Table 3 Numerical results for two subdomains with overlap equal to 1/4.
h2
1 16 1 32
1 32 1 64
1 64 1 128
Iter ERRh1 ERRh2 p1 p2 NDF h1 NDF h2
15 0:0015051 0:00245059 2:11598 2:05363 627 2405
15 0:000380237 0:00061632 2:0468 2:02249 2405 9417
15 9:88417E005 0:000157115 1:97408 1:98728 9417 37265
h1
5.1. Numerical experiments We have conducted numerical experiments on three different decompositions. 5.1.1. First domain decomposition X ¼ X1 [ X2 ; X1 ¼ 0; 98 ½0; 2; X2 ¼ 38 ; 2 ½0; 2. The size of the overlap region is
3 4
and results are listed in Table 1.
5.1.2. Second domain decomposition X ¼ X1 [ X2 ; X1 ¼ 0; 98 ½0; 2; X2 ¼ 58 ; 2 ½0; 2. The size of the overlap region is
1 2
and results are listed in Table 2.
5.1.3. Third domain decomposition X ¼ X1 [ X2 ; X1 ¼ 0; 98 ½0; 2; X2 ¼ 78 ; 2 ½0; 2. The size of the overlap region is
1 4
and results are listed in Table 3.
Remark 2. From the previous numerical results we can first see that the obtained convergence order is in agreement with the theory. We can also notice that, as the overlap size increases, the number of Schwarz iterations decreases.
6. Conclusion We have established an optimal convergence order for the finite element Schwarz alternating method for a linear elliptic PDE on two subdomains with nonmatching grids, combining a geometrical convergence of both the continuous and discrete corresponding Schwarz algorithms and standard finite element L1 – error estimate for linear elliptic PDEs. This approach is new and offers more practical perspectives than the one introduced in [8] as it enables us to control the error, on each
M. Boulbrachene, Q. Al Farei / Applied Mathematics and Computation 238 (2014) 21–29
29
subdomain, between the discrete Schwarz algorithm and the true solution. A future work will be devoted to a possible extension of this approach to linear parabolic PDEs. References [1] P.L. Lions, On the Schwarz alternating method I, in: R. Glowinski, G.H. Golub, G.A. Meurant, J. Periaux (Eds.), Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations, S.I.A.M., Philadelphia, 1988, ISBN 0-89871-220-3, pp. 1–42. [2] P.L. Lions, On the Schwarz alternating method II, in: T.F. Chan, R. Glowinski, J. Périaux, O.B. Widlund (Eds.), Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations, S.I.A.M., Philadelphia, 1989, ISBN 0-89871-233-5, pp. 47–70. [3] R. Glowinski, G.H. Golub, G.A. Meurant, J. Periaux, Domain Decomposition Methods for Partial Differential Equations, S.I.A.M., Philadelphia, 1988, ISBN 0-89871-220-3. [4] M. Kamranian, M. Dehghan, M. Tatari, An image denoising approach based on a meshfree method and the domain decomposition technique, Eng. Anal. Boundary Elem. 39 (2014) 101–110. [5] M. Dehghan, S.M. Molavi-Arabshahi, Comparison of preconditioning techniques for solving linear systems arising from the fourth order approximation of the three-dimensional elliptic equation, Appl. Math. Comput. 184 (2007) 156–172. [6] M. Dehghan, S.M. Molavi-Arabshahi, A simple form for the fourth order difference method for 3-D elliptic equations, Appl. Math. Comput. 184 (2007) 589–598. [7] J. Nitsche, L1 – convergence of finite element approximations, mathematical aspects of finite element methods, Lect. Notes Math. 606 (1977) 261–274. [8] M. Boulbrachene, P. Cortey-Dumont, J.C. Miellou, Mixing finite elements and finite differences on a subdomain decomposition method, in: T.F. Chan, R. Glowinski, J. Périaux, O.B. Widlund (Eds.), Proceedings of The First International Symp on Domain Decomposition Methods for Partial Differential Equations, S.I.A.M., Philadelphia, 1988, pp. 198–216. [9] X.-. Cai, T.P. Mathew, M. Sarkis, Maximum norm analysis of overlapping nonmatching grid discretization of elliptic equations, S.I.A.M J. Numer. Anal. 5 (2000) 1709–1728. [10] P.G. Ciarlet, P.A. Raviart, Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Eng. 2 (1973) 17– 31. [11] J. Karatson, S. Korotov, Discrete maximum principle for finite element solutions of nonlinear elliptic problems with mixed boundary conditions, Numer. Math. 99 (2005) 669–698.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Maximum norm error analysis of a nonmatching grids finite element method for linear elliptic PDEs q Messaoud Boulbrachene a,⇑, Qais Al Farei b a b
Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Muscat 123, Oman Higher College of Technology, IT Department, Muscat 123, Oman
a r t i c l e
i n f o
a b s t r a c t In this paper, we study a nonmatching grid finite element approximation of linear elliptic PDEs in the context of the Schwarz alternating domain decomposition.We show that the approximation converges optimally in the maximum norm, on each subdomain, making use of the geometrical convergence of both the continuous and corresponding discrete Schwarz sequences. We also give some numerical results to support the theory. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Elliptic PDEs Schwarz alternating method Nonmatching grids Finite element L1 – error estimate
1. Introduction The Schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. Extensive analysis of Schwarz alternating method for linear elliptic boundary value problems can be found in [1,2].The effectiveness of Schwarz methods for these problems, especially those in fluid mechanics, has been demonstrated in many papers. For that, we refer to the proceedings of the annual domain decomposition conference beginning with [3]. In this paper, we are interested in the error analysis in the maximum norm of a finite element Schwarz alternating method for linear elliptic problems on two overlapping subdomains with nonmatching grids: we consider a domain X which is the union of two overlapping subdomains where each subdomain has its own triangulation. This kind of discretization is very interesting as they can be applied to solving many practical problems which cannot be handled by global discretizations.They are earning particular attention of computational experts and engineers as they allow the choice of different mesh sizes and different orders of approximate polynomials in different subdomains according to the different properties of the solution and different requirements of the practical problems. For other kinds of discretization of elliptic PDEs, we refer to [4–6]. More specifically, we show that the approximation converges optimally on each subdomain, developing a new approach which consists of combining the geometrical convergence of both the continuous and discrete Schwarz sequences with an estimate in the maximum norm between the continuous and discrete Schwarz iterates. More precisely, if ðuni; Þ denotes the Schwarz sequence in the subdomain Xi and ðuni;hi Þ is its finite element counterpart with respect to the triangulation with meshsize hi , we show that
n ui uni;hi
L1 ðXi Þ
q
2
6 Ch jlog hj
The authors would like to thank the referee for his careful reading and valuable comments.
⇑ Corresponding author.
E-mail addresses: [email protected] (M. Boulbrachene), [email protected] (Q. Al Farei). http://dx.doi.org/10.1016/j.amc.2014.03.146 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
ð1:1Þ
22
M. Boulbrachene, Q. Al Farei / Applied Mathematics and Computation 238 (2014) 21–29
and
2 ui ui;h 1 6 Ch jlog hj i L ðX Þ i
ð1:2Þ
where ui denotes the true solution on Xi ; ui;hi is the uniform limit of the discrete Schwarz sequence ðuni;hi Þ, and C is a constant independent of n and h. For other works on maximum norm analysis of overlapping nonmatching grid methods for elliptic problems (cf. e.g., [8,9]). Now, we give an outline of the paper. In Section 2, we state the continuous Schwarz sequences and prove their geometrical convergence. In Section 3, we define their respective finite element counterparts in the context of nonmatching overlapping grids. Section 4 discusses the L1 - error analysis of the method, and Section 5 is devoted to some numerical experiments. 2. The continuous problem Consider the second order linear elliptic problem:
Du þ cu ¼ f
in X
u¼0
on @ X
ð2:1Þ
where X is a bounded polyhedral domain of R2 (or R3 Þ with boundary @ X; c a nonnegative function and f a given regular function. We decompose the domain X into two overlapping polyhedral subdomains X1 and X2 such that:
X ¼ X1 [ X2 and
X1 \ X2 – ; Let ci ¼ @ Xi \ Xj ; i; j ¼ f1; 2g; i – j denote the internal boundaries of Xi , and Ci ¼ @ Xi n ci the external boundaries such that i \ c j ¼ ;. Let also fi ¼ f =X . We associate with this problem the following Schwarz problem: find @ Xi ¼ Ci [ ci , and c i fu1 ; u2 g such that
8 > < Dui þ cui ¼ fi ui ¼ 0 > : ui ¼ uj
in Xi on Ci
ð2:2Þ
on ci
and
ui ¼ u=Xi Next, we define the associated Schwarz sequences and prove that they converge geometrically to the unique solution fu1 ; u2 g of system (2.2). 2.1. The continuous Schwarz sequences nþ1 Let u0 be an initial guess in CðXÞ where u02 ¼ u0 =X2 . We define the continuous Schwarz sequences ðunþ1 1 Þ in X1 and ðu2 Þ in X2 , by solving the following two subproblems:
8 nþ1 nþ1 > < Du1 þ cu1 ¼ f1 nþ1 u1 ¼ 0 > : nþ1 u1 ¼ un2
in X1 on C1 on c1
ð2:3Þ
and
8 nþ1 nþ1 > < Du2 þ cu2 ¼ f2 nþ1 u2 ¼ 0 > : nþ1 u2 ¼ unþ1 1
in X2 on C2
ð2:4Þ
on c2
respectively. Next, we shall prove that these Schwarz sequences converge geometrically to the solution of problem (2.2). For that let us define the following fixed point mapping. 2.1.1. A fixed point mapping associated with the Schwarz Method Indeed, given wj 2 CðXj Þ, let us first consider the following Dirichlet problem
M. Boulbrachene, Q. Al Farei / Applied Mathematics and Computation 238 (2014) 21–29
8 i þ cu i ¼ 0 in Xi > < Du on Ci ui ¼ 0 > : on ci ui ¼ wj
23
ð2:5Þ
For i; j 2 f1; 2g; i – j, let Ei ¼ Cðci Þ, we define the mappings:
T i : Ej # Ei
ð2:6Þ
v j # T i ðv j Þ where
i =c T i ðv j Þ ¼ u j
ð2:7Þ
Lemma 1 (see [8]). The mappings T i ; j ¼ 1; 2, are contractions, that is,
kT i kLðL1 ðcj Þ;L1 ðci ÞÞ ¼ qi < 1 Notation 1. From now and on, we shall adopt the following notations:
kkL1 ðX1 Þ ¼ kk1 ; kkL1 ðc1 Þ ¼ jj1 ;
kkL1 ðX2 Þ ¼ kk2 kkL1 ðc2 Þ ¼ jj2
pi ðv Þ ¼ v =ci 8v 2 CðXi Þ nþ1 Proposition 1. The sequences ðunþ1 1 Þ and ðu2 Þ; n P 0 produced by the Schwarz alternating method converge geometrically, in the maximum norm, to the solutions fu1 ; u2 g of problem (2.2) More precisely, we have
nþ1 u u1 1 6 qn1 qn2 u02 u2 2 1
ð2:8Þ
nþ1 0 u u u2 u2 2 6 qn1 qnþ1 2 2 2 2
ð2:9Þ
and
where
q ¼ max fq1 ; q2 g Proof. We prove this lemma by induction. Let us first prove (2.8). Indeed, subtracting (2.2) from (2.3) and (2.4), respectively, we get
8 nþ1 nþ1 > < Dðu1 u1 Þ þ cðu1 u1 Þ ¼ 0 in X1 unþ1 u1 ¼ 0 on C1 1 > : nþ1 n u1 u1 ¼ u2 u2 on c1 and
8 nþ1 nþ1 > < Dðu2 u2 Þ þ cðu2 u2 Þ ¼ 0 in X2 unþ1 u2 ¼ 0 on C2 2 > : nþ1 nþ1 u2 u2 ¼ u1 u1 on c2 n ¼ 1: Using both the definition of both T 1 and T 2 and the maximum principle, we have
2 u u1 6 u1 u2 6 T 1 ðp2 ðu1 u1 ÞÞ 6 q u1 u1 6 q T 2 ðp1 ðu0 u2 ÞÞ 6 q q u0 u2 1 1 1 1 2 2 1 2 1 2 1 1 1 2 2 1 n ¼ 2:
3 u u1 6 u2 u2 6 T 1 ðp2 ðu2 u1 ÞÞ 6 q1 u2 u1 6 q1 T 2 ðp1 ðu1 u2 ÞÞ 6 q1 q2 u1 u2 1 2 1 1 2 2 1 1 1 2 2 1 6 q1 q2 T 1 ðp2 ðu11 u1 ÞÞ1 6 q21 q2 u11 u1 2 6 q21 q2 T 2 ðp1 ðu02 u2 ÞÞ2 6 q21 q22 u02 u2 1
24
M. Boulbrachene, Q. Al Farei / Applied Mathematics and Computation 238 (2014) 21–29
So, by induction, we get
nþ1 u u1 1 6 qn1 qn2 u02 u2 1 1 Now, we can prove (2.9) in a similar way. Indeed, we have
1 u u2 6 u1 u1 6 T 2 ðp1 ðu0 u2 ÞÞÞ 6 q u0 u2 2 2 2 1 2 2 2 2 1 and
2 u u2 6 u2 u1 6 T 2 ðp1 ðu1 u2 ÞÞ 6 q2 u1 u2 6 q2 T 1 ðp2 ðu1 u1 ÞÞ 6 q1 q2 u1 u1 2 1 2 2 1 1 2 2 2 1 1 2 6 q1 q2 T 2 ðp1 ðu02 u2 ÞÞ2 6 q1 q22 u02 u2 1 Hence, we get
nþ1 0 u u u2 u2 2 6 qn1 qnþ1 2 2 2 1 Now, taking q ¼ maxðq1 ; q2 Þ, we get
nþ1 u u1 1 6 q2n u02 u2 2 1
and
nþ1 u u2 2 6 q2nþ1 u02 u2 2 2
3. Nonmatching grids discretization For i ¼ 1; 2, let shi be a standard quasi-uniform regular finite element triangulation in Xi ; hi being its mesh size. In order to construct an overlapping nonmatching grids discretization, we assume that the two triangulation are mutually independent on X1 \ X2 in the sense that a triangle belonging to one triangulation does not necessarily belong to the other one. To define the discrete Schwarz problems on overlapping nonmatching discretization, we introduce the finite element spaces
i Þ \ H1 ðXi Þ : v h j 2 P1 8K 2 shi g V hi ¼ f v h 2 C 0 ð X K 0 where V hi is the space of continuous piecewise linear functions, and P 1 is the space of polynomials with degree less than or equal to 1. Let pi be trace operator on ci and Ehi denote the range of pi defined on V hj . We also introduce the bilinear form:
Z
ai ðu; v Þ ¼
ðru:rv þ cuv Þ:dx
Xi
and the linear form
ðu; v Þ ¼
Z
u:v :dx
Xi
We, finally, denote by r hi , the usual interpolation operator on Ehi . The discrete maximum principle assumption: we assume that the matrices whose coefficients are ai ðul ; us Þ are Matrices [10,11]. (a matrix A is said to be an Matrix if its inverse A1 exists and satisfies A1 P 0). 3.1. The discrete problem We define the discrete analog of system (2.2) as follows: find fu1;h1 ; u2;h2 g such that
8 < ai ðui;hi ; v h Þ ¼ ðfi ; v h Þ 8 : ui;h ¼ uj;h i j
vh 2 Vh
i
ð3:1Þ
on ci
3.2. The discrete Schwarz sequences The corresponding discrete Schwarz problems are given by: find unþ1 1;h1 such that
8 8 < a1 ðunþ1 1;h1 ; v h Þ ¼ ðf1 ; v h Þ : unþ1 ¼ r ðp ðun ÞÞ 1 h1 2;h2 1;h1
vh 2 Vh
on c1
1
ð3:2Þ
25
M. Boulbrachene, Q. Al Farei / Applied Mathematics and Computation 238 (2014) 21–29
and find unþ1 such that 2;h2
(
a2 ðunþ1 8 2;h2 ; v h Þ ¼ ðf2 ; v h Þ nþ1 unþ1 2;h2 ¼ r h2 ðp2 ðu1;h1 ÞÞ
vh 2 Vh
2
ð3:3Þ
on c2
As in the continuous case, we are able to prove that the above Schwarz sequences converge geometrically, in the maxi mum norm, to the solutions u1;h1 ; u2;h2 of (3.1). For that, let us define the discrete analogue of problem (2.5): Given i;hi such that wj 2 CðXj Þ, find u
i;hi ; v h Þ ¼ 0 ai ðu 8 v h 2 V hi i;hi ¼ r hi ðpi ðwj ÞÞ on ci u
and the discrete fixed point mapping T hi
j Þ # Ehi T hi : Cðc
ð3:4Þ
v j # T h ðv j Þ i
where
T hi ð v i Þ ¼ r hi
pj ðui;hi Þ
ð3:5Þ
Lemma 2 [8]. Let the discrete maximum principle hold. Then, there exists a constant 0 < li < 1 independent of h such that
T h 1 i LðL ðc
jÞ ;L
1
ðci ÞÞ
6 li
nþ1 Proposition 2. Let Lemma 2 hold. Then the discrete sequences ðunþ1 1;h1 Þ; ðu2;h2 Þ converge geometrically, in the maximum norm, to the discrete solutions u1;h1 ; u2;h2 of (3.1), that is,
nþ1 n n u1;h1 u1;h1 6 l1 :l2 u02;h2 u2;h2 1
ð3:6Þ
2
and
nþ1 n nþ1 u2;h2 u2;h2 6 l1 l2 u02;h2 u2;h2 2
2
ð3:7Þ
Proof. The proof is similar to that of Proposition 1. Indeed, it suffices to use the definition of T hj and the discrete maximum principle. h
4. Maximum norm error Next, we shall estimate the error between the continuous Schwarz sequence and its discrete analogue. For that, we need to introduce the following auxiliary problems
(
~ nþ1 a1 ðu 8 1;h1 ; v h Þ ¼ ðf1 ; v h Þ
vh 2 Vh
1
vh 2 Vh
2
n ~ nþ1 u 1;h1 ¼ r h1 ðp1 ðu2 ÞÞ
ð4:1Þ
and
(
~ nþ1 a2 ðu 8 2;h2 ; v h Þ ¼ ðf2 ; v h Þ nþ1 ~ nþ1 u 2;h2 ¼ r h2 ðp2 ðu1 ÞÞ
ð4:2Þ
where un1 and un2 are the continuous Schwarz sequences. Lemma 3. Let h ¼ maxfh1 ; h2 g. Then we have nþ1 1l nþ1 2 Ch jlog hj u1 unþ1 1;h1 6 1 1l
ð4:3Þ
nþ2 1l nþ1 2 Ch jlog hj u2 unþ1 2;h2 6 2 1l
ð4:4Þ
and
where C is a constant independent of h and n.
26
M. Boulbrachene, Q. Al Farei / Applied Mathematics and Computation 238 (2014) 21–29
Proof. The proof will combine the discrete maximum principle and standard finite element maximum norm error estimate for linear PDEs (see [7]). Indeed, by subtracting problems (3.2) from (4.1) and (3.3) from (4.2), we respectively get
(
nþ1 ~ nþ1 a1 ðu 1;h1 u1;h1 ; v h Þ ¼ 0
~ nþ1 u 1;h1
unþ1 1;h1
¼ r h1 ð p
n 1 ðu2
un2;h2 ÞÞ
un1;h1 ÞÞ
8
vh 2 Vh
8
vh 2 Vh
1
and
(
nþ1 ~ nþ1 a1 ðu 2;h2 u2;h2 ; v h Þ ¼ 0
~ nþ1 u 2;h2
unþ1 2;h2
¼ r h2 ð p
n 2 ðu1
2
So, by induction, we have for [n ¼ 0: domain 1]
1 ~1 0 2 2 1 0 0 0 ~ 11;h u1 u11;h1 6 u11 u þ u 1;h1 u1;h1 6 Ch jlog hj þ r h1 ðp1 ðu2 u2;h2 ÞÞ 6 Ch jlog hj þ u2 u2;h2 1 1
1
1
2
2
1
2
2
6 Ch jlog hj þ Ch jlog hj 6 Ch jlog hj [n ¼ 0: domain 2]
1 ~1 2 1 1 1 ~ 12;h þ u 6 Ch log h þ r ð p ðu u ÞÞ u j j u2 u12;h2 6 u12 u 2 h 2;h 2;h 1 1;h 2 2 2 2 1 2 2 2 2 2 2 2 0 0 0 6 Ch jlog hj þ T h2 ðr h1 ðp1 ðu2 u2;h2 ÞÞ 6 Ch jlog hj þ lrh1 ðp1 ðu2 u02;h2 ÞÞ 6 Ch jlog hj þ lu02 u02;h2 2
2
2
1
2
2
6 Ch jlog hj þ l:Ch jlog hj 6 ð1 þ lÞCh jlog hj [n ¼ 1: domain 1]
2 ~2 2 2 2 1 1 1 1 ~ 21;h u1 u21;h1 6 u21 u þ u 1;h1 u1;h1 6 Ch jlog hj þ r h1 ðp1 ðu2 u2;h2 ÞÞ 6 Ch jlog hj þ p1 ðu2 u2;h2 Þ 1 1 1 1 1 1 2 2 2 6 Ch jlog hj þ T h1 ðr h2 ðp2 ðu11 u11;h1 ÞÞ 6 Ch jlog hj þ lrh2 ðp2 ðu11 u11;h1 Þ 6 Ch jlog hj þ lu11 u11;h1 1
2
2
2
1
2
6 Ch jlog hj þ lCh jlog hj 6 ð1 þ lÞCh jlog hj [n ¼ 1: domain 2]
2 ~2 2 2 2 2 2 1 1 ~ 22;h u2 u22;h2 6 u22 u þ u 2;h2 u2;h2 6 Ch jlog hj þ r h1 ðu1 u1;h1 Þ=c2 6 Ch jlog hj þ T h2 ðp2 ðu2 u2;h2 Þ 2 2 2 2 2 2 2 2 2 2 6 Ch jlog hj þ lph2 ðu12 u12;h2 Þ=c1 6 Ch jlog hj þ lu12 u12;h2 6 Ch jlog hj þ l½ð1 þ lÞCh jlog hj 1
2
2
2
6 ð1 þ l þ l ÞCh jlog hj [n ¼ 2; domain 1]
3 ~3 2 3 2 2 ~ 31;h þ u 6 Ch log h þ r ð p ðu u ÞÞ u j j u1 u31;h1 6 u31 u 1 h 1;h 1;h 2 2;h 1 1 1 1 2 1 1 1 1 2 2 2 2 2 2 6 Ch jlog hj þ T h1 ðr h2 ðp2 ðu1 u1;h1 ÞÞÞ 6 Ch jlog hj þ lr h2 ðp2 ðu1 u21;h1 ÞÞ 6 Ch jlog hj þ lu21 u21;h1 1
2
2
2
2
1
2
6 Ch jlog hj þ l½ð1 þ lÞCh jlog hj 6 ð1 þ l þ l ÞCh jlog hj [n ¼ 2: domain 2]
3 ~3 2 3 3 3 ~ 32;h u2 u32;h2 6 u32 u þ u 2;h2 u2;h2 6 Ch jlog hj þ r h2 ðp2 ðu1 u1;h1 ÞÞ 2 2 2 2 2 2 2 2 6 Ch jlog hj þ T h2 ðr h1 ðp1 ðu22 u22;h2 ÞÞÞ 6 Ch jlog hj þ lr h1 ðp1 ðu22 u22;h2 ÞÞ 6 Ch jlog hj þ lu22 u22;h2 2
2
2
2
1
2
3
2
6 Ch jlog hj þ l½ð1 þ l þ l ÞCh jlog hj 6 ð1 þ l þ l þ l ÞCh jlog hj So, by induction nþ1 1l nþ1 2 n 2 2 6 ð1 þ l þ l þ þ l ÞCh log h 6 Ch jlog hj j j u1 unþ1 1;h1 1 1l
and nþ2 1l nþ1 2 nþ1 2 2 ÞCh jlog hj 6 Ch jlog hj u2 unþ1 2;h2 6 ð1 þ l þ l þ þ l 2 1l
which completes the proof.
h
2
27
M. Boulbrachene, Q. Al Farei / Applied Mathematics and Computation 238 (2014) 21–29
Theorem 1. There exists a constant C independent of both n and h such that:
u1 u1;h 6 Ch2 jlog hj 1 1
ð4:5Þ
u2 u2;h 6 Ch2 jlog hj 2 1
ð4:6Þ
and
Proof. Let q ¼ max fq1 ; q2 g and l ¼ max fl1 ; l2 g. Then, using Propositions 1 and 2, and Lemma 3, we have nþ1 1l nþ1 2 2n 2n 0 u1 u1;h 6 u1 unþ1 þ unþ1 u2 u2 2 þ Ch jlog hj þ l u02;h u2;h unþ1 1 1 1;h þ u1;h u1;h 6 q 1 1 1 1 1l 2
So, by taking the limit, as n ! 1, we obtain
u1 u1;h 6 1
1 2 h jlog hj 1l
which yields (4.5). The proof of (4.6) is similar and will thus be omitted.
h
Remark 1. For computational purposes, it is interesting to estimate the error between the true solution and the discrete Schwarz sequence. Corollary 1. We have
2 ui uni;h 6 qn u02 u2 2 þ Ch jlog hj 1
Proof. Indeed,
ui uni;h 6 ui uni 2 þ uni unih 2 1
2 qn u02 u2 2 þ Ch jlog hj
5. Numerical experiments Our aim in this section is to validate the theoretical results of this paper. For that, let us consider the following elliptic problem:
Du ¼ f
in X ¼ ð0; 2Þ ð0; 2Þ
u¼0
on @ X
ð5:1Þ
where f ðx; yÞ ¼ 4ðx þ yÞ 2ðx2 þ y2 Þ is the forcing term and the exact solution is uExact ¼ xyðx 2Þðy 2Þ. We decompose the domain X into two overlapping rectangular subdomains X1 and X2 , supporting independent piecewise finite element triangulation; h1 is the mesh size of the triangulation in X1 , and h2 is the mesh size of the triangulation in X2 . cWe denote by Iter the number of Schwarz iterations on the subdomain Xi , by ERRhi the error between the exact solution and the Schwarz iterate on the subdomain Xi , i.e.,
Iter ERRhi ¼ uExact uIter i;hi ¼ uExact ui;hi i
L1 ðXi Þ
and by pi is the convergence order on the subdomain Xi , determined by the formula
pi ¼ 2
lnðERRhi Þ lnðERRh0i Þ ln ðNDFÞhi ln ðNDFÞh0
ð5:2Þ
i
where ðNDFÞhi refers to the number of degrees of freedom related to hi . The computation is carried out with a tolerance e ¼ 105 , that is
ERRhi 6 e:
28
M. Boulbrachene, Q. Al Farei / Applied Mathematics and Computation 238 (2014) 21–29 Table 1 Numerical results for two subdomains with overlap equal to 3/4.
h2
1 16 1 32
1 32 1 64
1 64 1 128
Iter ERRh1 ERRh2 p1 p2 NDF h1 NDF h2
7 0:000993274 0:00138449 2:11433 2:04483 627 3445
7 0:000248486 0:000346446 2:06139 2:02376 2405 13545
7 6:21738E005 8:66422E005 2:03001 2:01205 9417 53713
h1
Table 2 Numerical results for two subdomains with overlap equal to 1/2.
h2
1 16 1 32
1 32 1 64
1 64 1 128
Iter ERRh1 ERRh2 p1 p2 NDF h1 NDF h2
9 0:00114442 0:00194233 2:12042 2:04926 627 2925
9 0:000287017 0:000486318 2:05766 2:02542 2405 11481
9 7:21256E005 0:000121875 2:02369 2:0103 9417 45489
h1
Table 3 Numerical results for two subdomains with overlap equal to 1/4.
h2
1 16 1 32
1 32 1 64
1 64 1 128
Iter ERRh1 ERRh2 p1 p2 NDF h1 NDF h2
15 0:0015051 0:00245059 2:11598 2:05363 627 2405
15 0:000380237 0:00061632 2:0468 2:02249 2405 9417
15 9:88417E005 0:000157115 1:97408 1:98728 9417 37265
h1
5.1. Numerical experiments We have conducted numerical experiments on three different decompositions. 5.1.1. First domain decomposition X ¼ X1 [ X2 ; X1 ¼ 0; 98 ½0; 2; X2 ¼ 38 ; 2 ½0; 2. The size of the overlap region is
3 4
and results are listed in Table 1.
5.1.2. Second domain decomposition X ¼ X1 [ X2 ; X1 ¼ 0; 98 ½0; 2; X2 ¼ 58 ; 2 ½0; 2. The size of the overlap region is
1 2
and results are listed in Table 2.
5.1.3. Third domain decomposition X ¼ X1 [ X2 ; X1 ¼ 0; 98 ½0; 2; X2 ¼ 78 ; 2 ½0; 2. The size of the overlap region is
1 4
and results are listed in Table 3.
Remark 2. From the previous numerical results we can first see that the obtained convergence order is in agreement with the theory. We can also notice that, as the overlap size increases, the number of Schwarz iterations decreases.
6. Conclusion We have established an optimal convergence order for the finite element Schwarz alternating method for a linear elliptic PDE on two subdomains with nonmatching grids, combining a geometrical convergence of both the continuous and discrete corresponding Schwarz algorithms and standard finite element L1 – error estimate for linear elliptic PDEs. This approach is new and offers more practical perspectives than the one introduced in [8] as it enables us to control the error, on each
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subdomain, between the discrete Schwarz algorithm and the true solution. A future work will be devoted to a possible extension of this approach to linear parabolic PDEs. References [1] P.L. Lions, On the Schwarz alternating method I, in: R. Glowinski, G.H. Golub, G.A. Meurant, J. Periaux (Eds.), Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations, S.I.A.M., Philadelphia, 1988, ISBN 0-89871-220-3, pp. 1–42. [2] P.L. Lions, On the Schwarz alternating method II, in: T.F. Chan, R. Glowinski, J. Périaux, O.B. Widlund (Eds.), Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations, S.I.A.M., Philadelphia, 1989, ISBN 0-89871-233-5, pp. 47–70. [3] R. Glowinski, G.H. Golub, G.A. Meurant, J. Periaux, Domain Decomposition Methods for Partial Differential Equations, S.I.A.M., Philadelphia, 1988, ISBN 0-89871-220-3. [4] M. Kamranian, M. Dehghan, M. Tatari, An image denoising approach based on a meshfree method and the domain decomposition technique, Eng. Anal. Boundary Elem. 39 (2014) 101–110. [5] M. Dehghan, S.M. Molavi-Arabshahi, Comparison of preconditioning techniques for solving linear systems arising from the fourth order approximation of the three-dimensional elliptic equation, Appl. Math. Comput. 184 (2007) 156–172. [6] M. Dehghan, S.M. Molavi-Arabshahi, A simple form for the fourth order difference method for 3-D elliptic equations, Appl. Math. Comput. 184 (2007) 589–598. [7] J. Nitsche, L1 – convergence of finite element approximations, mathematical aspects of finite element methods, Lect. Notes Math. 606 (1977) 261–274. [8] M. Boulbrachene, P. Cortey-Dumont, J.C. Miellou, Mixing finite elements and finite differences on a subdomain decomposition method, in: T.F. Chan, R. Glowinski, J. Périaux, O.B. Widlund (Eds.), Proceedings of The First International Symp on Domain Decomposition Methods for Partial Differential Equations, S.I.A.M., Philadelphia, 1988, pp. 198–216. [9] X.-. Cai, T.P. Mathew, M. Sarkis, Maximum norm analysis of overlapping nonmatching grid discretization of elliptic equations, S.I.A.M J. Numer. Anal. 5 (2000) 1709–1728. [10] P.G. Ciarlet, P.A. Raviart, Maximum principle and uniform convergence for the finite element method, Comput. Methods Appl. Mech. Eng. 2 (1973) 17– 31. [11] J. Karatson, S. Korotov, Discrete maximum principle for finite element solutions of nonlinear elliptic problems with mixed boundary conditions, Numer. Math. 99 (2005) 669–698.
