Résumé: With the development of parallel computers, domain decomposition methods(DDM) have been increasingly used as important tools for solving boundary value problems. There exists, in practice, two ideas of decomposition of the domain: with and without overlapping of subdomains. This work is concerned with the analysis of the generalized overlapping (Schwarz) DDM, by using Robin boundary conditions on the interfaces. The nonoverlapping case was studied in [2,3,7,10,11]. We are interested in the study of the convergence of the iteratif process in the continuous and discrete cases. We use an energy method of Lions [7] to prove the convergence of the iteratif process and a generalization of a relaxation procedure first used by Deng [2], to avoid the computation of normal derivatives and to facilitate the application of this method to discrete problems and to get an optimal convergence rate.

Résumé: In this work we present an improvement of Steffensen's method for computing numerical approximation of nonlinear equations f(x)=0. The order of convergence of this new iterative method (with two-steps) is p2, knowing that the method of Steffensen (with only one step) is of order 2p-1 .

Résumé: A posteriori error estimates for the generalized overlapping domain decomposition method GODDM i.e., with Robin boundary conditions on the interfaces, for second order boundary value problems, are derived. We show that the error estimate in the continuous case depends on the differences of the traces of the subdomain solutions on the interfaces. After discretization of the domain by finite elements we use the techniques of the residual a posteriori error analysis to get an a posteriori error estimate for the discrete solutions on subdomains. The results of some numerical experiments are presented to support the theory.

Résumé: In this paper, we analyse the ‘defect-correction’ technique on a general smooth region, via composite finite-element meshes (a Cartesian mesh and a polar mesh) on two overlapping subdomains (a rectangle and an annulus). Boundary interpolatory mappings of higher degree are used, in the Schwarz method, to pass from one mesh to another. An explicit relation is given between the degree of these mappings and the number of optimal corrections to be computed. Optimal convergence results for the discrete bilinear basic solution, in higher-order discrete Sobolev norms, are obtained on the subdomains. Because the success of the defect-correction technique is based on the uniformity of the discretization and the regularity of the exact solution, the defects are computed on the subdomains in the same way as for the basic solution. Optimal O(h2) improvement per correction is obtained. Numerical results are presented to support the theory. Keywords: defect correction; Schwarz method; composite meshes; bilinear finite elem

Résumé: Highly accurate approximation is obtained through the techniques of defect correction and domain decomposition for second-order elliptic boundary value problems on a disc. The basic solution is computed using the Schwarz domain decomposition procedure and bilinear Galerkin 4nite element approximation on each subdomain to get an O(h2) accurate basic solution in higher-order discrete Sobolevnorms. The defects are then computed using high-order polynomials (Lagrange polynomials or splines) to get as many O(h2) corrections as possible. c 2001 Elsevier Science B.V. All rights reserved. MSC: 65N30; 65N55; 65B05; 65J10 Keywords: Defect correction; Schwarz domain decomposition; Bilinear 4nite elements; Elliptic problems

Résumé: Highly accurate approximation is obtained through the techniques of defect correction and domain decomposition for second-order elliptic boundary value problems on a disc. The basic solution is computed using the Schwarz domain decomposition procedure and bilinear Galerkin 4nite element approximation on each subdomain to get an O(h2) accurate basic solution in higher-order discrete Sobolevnorms. The defects are then computed using high-order polynomials (Lagrange polynomials or splines) to get as many O(h2) corrections as possible. c
2001 Elsevier Science B.V. All rights reserved. MSC: 65N30; 65N55; 65B05; 65J10 Keywords: Defect correction; Schwarz domain decomposition; Bilinear 4nite elements; Elliptic problems

Résumé: The defect correction technique, based on the Galerkin finite element method, is analyzed as a procedure to obtain highly accurate numerical solutions to second‐order elliptic boundary value problems. The basic solutions, defined over a rectangular region Ω, are computed using continuous piecewise bilinear polynomials on rectangles. These solutions are O(h2) accurate globally in the second‐order discrete Sobolev norm. Corrections to these basic solutions are obtained using higher‐order piecewise polynomials (Lagrange polynomials or splines) to form defects. An O(h2) improvement is gained on the first correction. The lack of regularity of the discrete problems (beyond the second‐order Sobolev norm) makes it impossible to retain this order of improvement, but for problems satisfying certain periodicity conditions, straightforward arbitrary accuracy is obtained, since these problems possess high‐order regularity. © 1992 John Wiley & Sons, Inc.

Résumé: With the development of parallel computers, domain decomposition methods(DDM) have been increasingly used as important tools for solving boundary value problems. There exists, in practice, two ideas of decomposition of the domain: with and without overlapping of subdomains. This work is concerned with the analysis of the generalized overlapping (Schwarz) DDM, by using Robin boundary conditions on the interfaces. The nonoverlapping case was studied in [2,3,7,10,11]. We are interested in the study of the convergence of the iteratif process in the continuous and discrete cases. We use an energy method of Lions [7] to prove the convergence of the iteratif process and a generalization of a relaxation procedure first used by Deng [2], to avoid the computation of normal derivatives and to facilitate the application of this method to discrete problems and to get an optimal convergence rate.

Résumé: In this work we present an improvement of Steffensen's method for computing numerical approximation of nonlinear equations f(x)=0. The order of convergence of this new iterative method (with two-steps) is p2, knowing that the method of Steffensen (with only one step) is of order 2p-1 .