Abstract
Hereafter, we describe and analyze, from both a theoretical and a numerical point of view, an iterative method for efficiently solving symmetric elliptic problems with possibly discontinuous coefficients. In the following, we use the Preconditioned Conjugate Gradient method to solve the symmetric positive definite linear systems which arise from the finite element discretization of the problems. We focus our interest on sparse and efficient preconditioners. In order to define the preconditioners, we perform two steps: first we reorder the unknowns and then we carry out a (modified) incomplete factorization of the original matrix. We study numerically and theoretically two preconditioners, the second preconditioner corresponding to the one investigated by Brand and Heinemann [2]. We prove convergence results about the Poisson equation with either Dirichlet or periodic boundary conditions. For a meshsizeh, Brand proved that the condition number of the preconditioned system is bounded byO(h −1/2) for Dirichlet boundary conditions. By slightly modifying the preconditioning process, we prove that the condition number is bounded byO(h −1/3).
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Communicated by c. Brezinski
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Ciarlet, P. Repeated Red-Black ordering: a new approach. Numer Algor 7, 295–324 (1994). https://doi.org/10.1007/BF02140688
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DOI: https://doi.org/10.1007/BF02140688