MiS Preprint Repository

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 ($\mathcal{H}$-matrix) formats we elaborate the approximate Schur complement inverse in an explicit form. The required cost $\mathcal{O}(N_{\mathrm{\Gamma }}{\log }^q{N}_{\mathrm{\Gamma }})$ is almost linear in $N_{\mathrm{\Gamma }}$ -- the number of degrees of freedom on the interface. As input, we require the Schur complement matrices corresponding to subdomains and represented in the $\mathcal{H}$-matrix format. In the case of piecewise constant coefficients these matrices can be computed via the BEM representation with the cost $\mathcal{O}(N_{\mathrm{\Gamma }}{\log }^q{N}_{\mathrm{\Gamma }})$, while in the general case the FEM discretisation leads to the complexity $O(N_{\mathrm{\Omega }}{\log }^q{N}_{\mathrm{\Omega }})$, where $N_{\mathrm{\Omega }}$ is the number of degrees of freedom in the domain.