Direct Schur Complement Method by Hierarchical Matrix Techniques

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(NΓ log q NΓ ) 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 log q NΓ ), while in the general case the FEM discretisation leads to the complexity O(NΩ log q NΩ).