Domain Decomposition Methods for Elliptic Problems with High Contrast Coefficients Revisited

2022/12/23 by Na, Xuyang, Xu, Xuejun
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2212.12216

Abstract

In this paper, we revisit the nonoverlapping domain decomposition methods for solving elliptic problems with high contrast coefficients. Some interesting results are discovered. We find that the Dirichlet-Neumann algorithm and Robin-Robin algorithms may make full use of the ratio of coefficients. Actually, in the case of two subdomains, we show that their convergence rates are O(ε), if ν₁≪ν₂, where ε= ν₁/ν₂ and ν₁,ν₂ are coefficients of two subdomains. Moreover, in the case of many subdomains, the condition number bounds of Dirichlet-Neumann algorithm and Robin-Robin algorithm are 1+ε(1+log(H/h))² and C+ε(1+log(H/h))², respectively, where ε may be a very small number in the high contrast coefficients case. Besides, the convergence behaviours of the Neumann-Neumann algorithm and Dirichlet-Dirichlet algorithm may be independent of coefficients while they could not benefit from the discontinuous coefficients. Numerical experiments are preformed to confirm our theoretical findings.

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