Optimized global-in-time Schwarz algorithm for diffusion equations with discontinuous and spatially variable coefficients

Florian Lemarié 1, * Laurent Debreu 1 Eric Blayo 1
* Corresponding author
1 MOISE - Modelling, Observations, Identification for Environmental Sciences
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
Abstract : In this report we present a global-in-time non-overlapping Schwarz method applied to the one dimensional unsteady diffusion equation. We address specifically the problem with discontinuous diffusion coefficients, our approach is therefore especially designed for subdomains with heterogeneous properties. We derive efficient interface conditions by solving analytically the minmax problem associated to the search of optimized conditions in a Robin-Neumann case and in a two-sided Robin-Robin case with constant coefficients. We study the impact of the finiteness of the subdomains on the optimized conditions. Then we address the problem with spatially variable coefficients. We derive a new approach to determine the convergence factor of the algorithm, which enables to optimize the convergence speed. The theoretical results are illustrated by numerical experiments in the case of Robin-Robin and Dirichlet-Neumann interface conditions.
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Florian Lemarié, Laurent Debreu, Eric Blayo. Optimized global-in-time Schwarz algorithm for diffusion equations with discontinuous and spatially variable coefficients. [Research Report] RR-6663, INRIA. 2008, pp.45. ⟨inria-00324533⟩

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