In this paper, we present a family of domain decomposition based on an Aitken-like acceleration of the Schwarz method seen as an iterative procedure with a linear rate of convergence. This paper is a generalization of the method first introduced at the 12th International Conference on Domain Decomposition that was restricted to regular Cartesian grids. The potential of this method to provide scalable parallel computing on a geographically broad grid of parallel computers was demonstrated for some linear and nonlinear elliptic problems discretized by finite differences on a Cartesian mesh. The main purpose of this paper is to present a generalization of the method to nonuniform Cartesian meshes. The salient feature of the method consists of accelerating the sequence of traces on the artificial interfaces generated by the Schwarz procedure using a good approximation of the main eigenvectors of the trace transfer operator. For linear separable elliptic operators, our solver is a direct solver. For nonlinear operators, we use an approximation of the eigenvectors of the Jacobian of the trace transfer operator. The acceleration is then applied to the sequence generated by the Schwarz algorithm applied directly to the nonlinear operator. Introduction.The classical Schwarz method [28] has been extensively analyzed in the past; see, for example, the book [29] and its references. This method was first devised as a tool to prove existence and uniqueness results for elliptic problems. Nowadays the Schwarz method is used as an iterative domain decomposition (DD) method or a preconditioner for a Krylov method. The classical Schwarz iterative scheme that uses Dirichlet boundary conditions at the artificial interfaces between overlapping subdomains has very slow numerical convergence. To accelerate the convergence optimized boundary conditions (OBCs) can replace the Dirichlet boundary conditions. Various OBCs have been proposed and analyzed by a number of authors. We refer to [1,12,25] for a recent survey of these methods. The Aitken-like acceleration is an alternative method that consists of postprocessing the sequence of interfaces generated by the iterative DD solver and might be applied to any subdomainwise relaxation scheme including those with OBCs.The idea of using Aitken acceleration [19,30] on the classical additive Schwarz DD method [22,23,24,28,29] was introduced in [15]. These authors have called the corresponding method the Aitken-Schwarz (AS) method. They have shown its great numerical performances on linear and nonlinear elliptic problems discretized by a five-point scheme on a rectangular Cartesian grid [18]. More recently, it was shown that this technique gives efficient metacomputing of the Poisson and Bratu problem in three space dimensions with a broad network of supercomputers linked

No abstract

No abstract

Summary. (and Introduction) In this paper, we present a family of domain decomposition based on Aitken like acceleration of the Schwarz method seen as an iterative procedure with linear rate of convergence. This paper is a generalization of the method first introduced in Garbey and Tromeur-Dervout [2001] that was restricted to Cartesian grids. The general idea is to construct an approximation of the eigenvectors of the trace transfer operator associated to dominant eigenvalues and accelerate these components after few Schwarz iterates. We consider here examples with the finite volume approximation on general quadrangle meshes of Faille [1992] and finite element discretization.