Direct Schur complement method by hierarchical matrix techniques

Abstract

The goal of this paper is the construction of a data-sparse approximation to the Schur complement on the interface corresponding to FEM and BEM approximations of an elliptic equation by domain decomposition. Using the hierarchical (H-matrix) formats we elaborate the approximate Schur complement inverse in an explicit form. The required cost O(NΓlogqNΓ) is almost linear in NΓ - the number of degrees of freedom on the interface. As input, we use the Schur complement matrices corresponding to subdomains and represented in the H-matrix format. In the case of piecewise constant coefficients these matrices can be computed via the BEM representation with the cost O(NΓ logq NΓ), while in the general case the FEM discretisation leads to the complexity O(NΩ logqNΩ).