Improved interface conditions for 2D domain decomposition with corners: a theoretical determination

Abstract

This article deals with a local improvement of domain decomposition methods for 2-dimensional elliptic problems for which either the geometry or the domain decomposition presents conical singularities. The problem amounts to determining the coefficients of interface boundary conditions so that the domain decomposition algorithm has rapid convergence. Specific problems occur in the presence of conical singularities. Starting from the method used for regular interfaces, we derive a local improvement by matching the singularities, that is, the initial terms of the asymptotic expansion arond the corner, provided by Kondratiev theory. This theoretical approach leads to the explicit computation of coefficients in the interface boundary conditions, which have been tested numerically. This final numerical step is presented in a companion article [4]. This article focuses on the method used to compute these coefficients and provides detailed examples for a model problem.