Domain Decomposition Methods for Elliptic Problems with High Contrast Coefficients Revisited
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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 ν_1≪ν_2, where ϵ = ν_1/ν_2 and ν_1,ν_2 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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