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On the convergence of algebraic optimizable Schwarz methods with applications to elliptic problems
, 2007
"... Abstract. The Schwarz method can be used for the iterative solution of elliptic boundary value problems on a large domain Ω. One subdivides Ω into smaller, more manageable, subdomains and solves the differential equation in these subdomains using appropriate boundary conditions. SchwarzRobin method ..."
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Abstract. The Schwarz method can be used for the iterative solution of elliptic boundary value problems on a large domain Ω. One subdivides Ω into smaller, more manageable, subdomains and solves the differential equation in these subdomains using appropriate boundary conditions. SchwarzRobin methods use Robin conditions on the artificial interfaces for information exchange at each iteration. Optimized Schwarz Methods (OSM) are those in which one optimizes the Robin parameters. While the convergence theory of classical Schwarz methods (with Dirichlet conditions on the artificial interface) is well understood, the overlapping SchwarzRobin methods still lack a complete theory. In this paper, an abstract Hilbert space version of the OSM is presented, together with an analysis of conditions for its convergence. It is also shown that if the overlap is relatively uniform, these convergence conditions are met for SchwarzRobin methods for twodimensional elliptic problems, for any positive Robin parameter. In the discrete setting, we obtain that the convergence rate ω(h) varies like a polylogarithm of h. Numerical experiments show that the methods work well and that the convergence rate does not appear to depend on h. 1. Introduction. Schwarz
An optimal block iterative method and preconditioner for banded matrices with applications to PDEs on irregular domains
, 2010
"... Abstract. Classical Schwarz methods and preconditioners subdivide the domain of a partial differential equation into subdomains and use Dirichlet transmission conditions at the artificial interfaces. Optimized Schwarz methods use Robin (or higher order) transmission conditions instead, and the Robin ..."
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Abstract. Classical Schwarz methods and preconditioners subdivide the domain of a partial differential equation into subdomains and use Dirichlet transmission conditions at the artificial interfaces. Optimized Schwarz methods use Robin (or higher order) transmission conditions instead, and the Robin parameter can be optimized so that the resulting iterative method has an optimized convergence factor. The usual technique used to find the optimal parameter is Fourier analysis; but this is only applicable to certain regular domains, for example, a rectangle, and with constant coefficients. In this paper, we present a completely algebraic version of the optimized Schwarz method, including an algebraic approach to find the optimal operator or a sparse approximation thereof. This approach allows us to apply this method to any banded or block banded linear system of equations, and in particular to discretizations of partial differential equations in two and three dimensions on irregular domains. With the computable optimal operator, we prove that the optimized Schwarz method converges in no more than two iterations, even for the case of many subdomains (which means that this optimal operator communicates globally). Similarly, we prove that when we use an optimized Schwarz preconditioner with this optimal operator, the underlying minimal residual Krylov subspace method (e.g., GMRES) converges in no more than two iterations. Very fast convergence is attained even when the optimal transmission operator is approximated by a sparse matrix. Numerical examples illustrating these results are presented. AMS subject classifications. 65F08, 65F10, 65N22, 65N55
AN OPTIMAL BLOCK ITERATIVE METHOD AND PRECONDITIONER FOR BANDED MATRICES WITH APPLICATIONS TO PDES ON IRREGULAR DOMAINS∗
, 2010
"... An optimal block iterative method and ..."
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