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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 893182, 21 pages
Maximum Norm Analysis of an Arbitrary Number of Nonmatching Grids Method for Nonlinears Elliptic PDES
Department of Mathematics, Laboratory of Applied Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, El Hadjar, 23000 Annaba, Algeria
Received 11 August 2013; Revised 7 October 2013; Accepted 9 October 2013
Academic Editor: Srinivasan Natesan
Copyright © 2013 Abida Harbi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We provide a maximum norm analysis of a finite element Schwarz alternating method for a nonlinear elliptic PDE on an arbitrary number of overlapping subdomains with nonmatching grids. We consider a domain which is the union of an arbitrary number of overlapping subdomains where each subdomain has its own independently generated grid. The meshes being mutually independent on the overlap regions, a triangle belonging to one triangulation does not necessarily belong to the other ones. Under the a Lipschitz assumption on the nonlinearity, we establish, on each subdomain, an optimal error estimate between the discrete Schwarz sequence and the exact solution of the PDE.
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 result from solving a sequence of elliptic boundary value problems in each of the subdomains. The effectiveness of the Schwarz methods for various classes of nonlinear elliptic PDE problems has been demonstrated in many papers; see [1–4] and the references therein. Also the effectiveness of the Schwarz methods for these problems, especially those in fluid mechanics, has been demonstrated in many papers. See proceedings of the annual domain decomposition conference beginning with .
In [6, 7], an optimal convergence order is obtained for nonlinear elliptic PDE, in the context of two overlapping nonmatching grids, in the sense that each subdomain has its own independent discretization by finite element method. This kind of discretization is very interesting as it can be applied to solve many practical problems which cannot be handled by global discretizations. Discretizations 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. Quite a few works on maximum norm error analysis of overlapping nonmatching grids methods for elliptic problems are known in the literature; compare and confer, for example, [8–10].
This paper is a continuation of previous work , attempting to generalize the obtained result related to convergence order also in the context of nonmatching grids in domain which consists of a union of an arbitrary number of subdomains. It is proved that the error estimate remains true also for more than two subdomains.
The proof of the main result which consists of estimating the error in the maximum norm between the continuous solution of the problem and the discrete Schwarz iterates stands on a Lipschitz continuous dependency with respect to both the boundary condition and the source term for linear elliptic equations. The optimal convergence order is then derived making use of standard finite elementerror estimate for linear elliptic equations.
Now, we give an outline of the paper. In Section 2 we state the continuous alternating Schwarz sequences and define their respective finite element counterparts in the context of nonmatching overlapping grids. Section 3 is devoted to theerror analysis of the method.
We begin by laying down some definitions and classical results related to linear elliptic equations.
2.1. Linear Elliptic Equations
Letbe a bounded polyhedral domain oforwith sufficiently smooth boundary. We consider the following: the bilinear form the linear form the right hand side the space whereis a regular function defined on.
We consider the linear elliptic equation: Findsuch that where, such that
Letbe the space of finite elements consisting of continuous piecewise linear functionsvanishing on, and letbe the basis function of. The discrete counterpart of (5) consists of findingsuch that where andis an interpolation operator on.
Lemma 2 (cf. ). Letsatisfynonnegative, andon. Thenon.
The proposition below establishes a Lipschitz continuous dependency of the solution with respect to the data.
Notation. Let;be a pair of data and;the corresponding solutions to (5).
Proposition 3 (cf. ). Under conditions of the preceding lemma, one has:
Remark 4. Lemma 2 stays true in the discrete case.
Indeed, assume that the discrete maximum principle (d.m.p) holds; that is, the matrix resulting from the finite element discretization is an M-Matrix. Then one has the following.
Lemma 5. Letsatisfyandon. Then,on.
Proof. The proof is a direct consequence of the discrete maximum principle.
Let;be a pair of data and;the corresponding solutions to (7).
2.2. The Schwarz Alternating Methods for Nonlinear PDEs
Consider the nonlinear PDE: or in its weak form whereis a nondecreasing nonlinearity. We assume thatis a Lipschitz continuous on, that is such that whereis the constant defined in (6).
We suppose that the problem (12) has a subsolutionand aersolutionwhich satisfyDefine the sector of smooth functions as follows: Finally we assume that where With these assumptions cf. , the problem (12) has a unique solution in the sectorWe can easily see that (12) is equivalent to whereand we note that the functionalsatisfies (17) and (18). We introduce the following comparison lemma.
Lemma 7 (cf. ). Suppose thatandonwith bothpositive inandonThenon
We decomposeintooverlapping smooth subdomainssuch that We denote bythe boundary ofand the interior boundaries by We assume that the intersection ofand,is empty. Let We associate problem (13) with the following system. Findsolution of where and (cf. )
In the sequel, we define a subsolution and a supersolution of (23)
2.3. The Continuous Schwarz Sequences
Letbe an initialization data:
We define the alternating Schwarz sequencesonsuch that each term of each sequence,solves where (cf. )
Theorem 9. The Schwarz sequences,converge to the solution of (23).
Proof. We first show by induction, that each term of the Schwarz sequences is well defined inIndeed, for, we begin by subdomain one, since,andsatisfy
Lemma 7 implies that
In subdomain 2, the condition of the interior boundaryis given by
so (37) implies that
On the other hand,, andsatisfy, respectively,
Then Lemma 7 implies that
The same idea is used sequentially in the remainder of the proof related to the rest of subdomains,. Indeed we obtain forand for
Now, let us assume that
and prove that
Indeed, in subdomain 1 (43) implies that
Since then Lemma 7 implies that Using (47) and (43), we can write We can easily obtain using (48) and by adopting the same approach used in subdomain one. The result related to rest of subdomains,, is obtained sequentially by similar way; that is
We demonstrate, by induction, that the Schwarz sequences are nondecreasing. Indeed forwe have demonstrated in (50) that, for,
Now, let us assume that and prove that
Begining by subdomain one, (52) implies that Making use of (54), We get by using Lemma 7 On the other hand, (52) and (56) imply then making use of (57), We get by using Lemma 7 the following result related to the second subdomain: The same idea is sequentially applied to the rest of subdomains. Finally, we obtain
The sequencesare bounded and nondecreasing, so they are monotone converging pointwise to,. By using an elliptic regularity argument, we can see that functionssatisfy the same PDE oncf. [3, 12].
2.4. The Discretization
Forletbe a standard regular and quasiuniform finite element triangulation in;being the meshsize. Themeshes is mutually independent onwhere; that is a triangle belonging to one triangulation does not necessarily belong to the other ones.
We consider the following discrete spaces: and for every, we set wheredenotes the interpolation operator on.
Note that as the meshesare independent over the overlapping subdomains, it is impossible to formulate a global approximate problem which would be the direct discrete counterpart of problem (13).
2.5. The Discrete Schwarz Sequences
Now, we define the discrete counterparts of the continuous Schwarz sequences defined in (30). Indeed, letbe the discrete analog of, defined in (29) and we define forthe discrete sequencesonsuch thatsolves where
Notation. From now on, we will adopt the following notations:
3. Error Analysis
3.1. The Auxiliary Schwarz Problems
This section is devoted to the proof of the main result of the present paper. To that end, we begin by introducingdiscrete auxiliary problems.
We define the following problems; forsolves It is then clear thatis the finite element approximation ofsolutions of (23). Therefore, making use of standard maximum norm estimates for linear elliptic problems, we have whereis a constant independent of.
3.1.1. The Main Results
Theorem 10. Let. Then, there exists a constantindependent of bothandsuch that
Proof. The proof of (68) will be carried by induction and is decomposed in three principal parts where each part is devoted to one of the following situations: situation (A), situation (B), and situation (C), respectively. The three situations are defined by
and in the last situation we assume that a nonempty subsetof the setexists such that
To this end, we apply Proposition 6 and Theorem 1 for each subdomainas follows:
The last inequality implies that
The intervalwhichbelongs to is divided into two subintervals as follows:
and in each situation (A), (B), or (C) we deal with each subinterval separately.
Part 1. We consider situation (A). We begin the proof by the first subinterval that is, so Forapplying (73) to subdomain one, we get We have to distinguish between two cases: or Case 1 implies that Then, Case 2 implies that By multiplying (81) bywe get Sois bounded by bothandthen or That is, or which implies that or It is clear that both cases (a) and (b) are true because both coincide with (69). So for either of them there is a contradiction, and thus case 2 is impossible or case 2 is possible only if Then case 2 implies that Hence in both cases 1 and 2, we obtain Similarly applying (73) to the second subdomain, we get We have to distinguish the three following cases or or Case 1 implies that Then, We note that the two last inequalities coincide with (91) and (69), respectively. Case 2 implies that By multiplying (98) by, we get We remark thatis bounded by bothandthen or Which implies that or That is or It is clear that both cases (a) and (b) are true because they coincide with (91). So there is either contradiction and case 2 is impossible or cases 2 is possible and we must have Then case 2 implies With which coincides with (69). Case 3 implies By multiplying (109) bywe get Sois bounded by bothandthen w have or Which implies or
That is, or It is clear that both cases (c) and (d) are true because they coincide with (69). So there is contradiction and case 3 is impossible or case 3 is possible only if Then case 3 implies that with Hence, in the three cases 1, 2, and 3 we get Equations (91) and (120) imply Similarly applying the same idea for the rest of subdomain, orderly, we get Now, let us assume that and prove that
Equation (124) is obtained sequentially in the order of the numbering of subdomains. We begin by subdomain 1; indeed applying (73) to subdomain one, for iteration, we get
We have to distinguish the two following cases or Case 1 implies Then Case 2 implies By multiplying (130) bywe get Sois bounded by bothandthen or That is or Which implies or It is clear that both cases (a) and (b) are true because both coincide with (123). So either there is a contradiction and thus case 2 is impossible or case 2 is possible only if Then case 2 implies Hence in both cases 1 and 2 we obtain Similarly applying (73) to the second subdomain, for iterationwe get We have to distinguish the three following cases or or Case 1 implies Then We note that the two last inequalities coincide with (140) and (123) respectively. Case 2 implies that By multiplying (147) bywe get We remark thatis bounded by bothandthen or Which implies or That is