Domain Decomposition Methods for Elliptic Problems with High Contrast Coefficients Revisited

TL;DR

This paper analyzes and revisits domain decomposition methods for elliptic problems with high contrast coefficients, revealing their convergence behaviors and condition number bounds, and confirming findings through numerical experiments.

Contribution

It provides new insights into the convergence rates and condition number bounds of various domain decomposition algorithms for high contrast elliptic problems.

Findings

Dirichlet-Neumann and Robin-Robin algorithms leverage coefficient ratios.

Convergence rate for two subdomains is $O(rac{ u_1}{ u_2})$ when $ u_1 \

Condition number bounds depend on the contrast ratio $rac{ u_1}{ u_2}$ and mesh parameters.

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 $O(\epsilon)$, if $\nu_1\ll\nu_2$, where $\epsilon = \nu_1/\nu_2$ and $\nu_1,\nu_2$ 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 $1+\epsilon(1+\log(H/h))^2$ and $C+\epsilon(1+\log(H/h))^2$, respectively, where $\epsilon$ 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.