Convergence of the multiplicative Schwarz method for singularly perturbed convection-diffusion problems discretized on a Shishkin mesh

We analyze the convergence of the multiplicative Schwarz method applied to nonsymmetric linear algebraic systems obtained from discretizations of one-dimensional singularly perturbed convection-diffusion equations by upwind and central finite differences on a Shishkin mesh. Using the algebraic structure of the Schwarz iteration matrices we derive bounds on the infinity norm of the error that are valid from the first step of the iteration. Our bounds for the upwind scheme prove rapid convergence of the multiplicative Schwarz method for all relevant choices of parameters in the problem. The analysis for the central difference is more complicated, since the submatrices that occur are nonsymmetric and sometimes even fail to be M -matrices. Our bounds still prove the convergence of the method for certain parameter choices.

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