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MiS Preprint
25/2004
Direct Schur complement method by domain decomposition based on $\mathcal{H}$-matrix approximation
Wolfgang Hackbusch, Boris N. Khoromskij and Ronald Kriemann
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 ($\mathcal{H}$-matrix) formats we elaborate the approximate Schur complement inverse in an explicit form. The required cost $\mathcal{O}(N_{\Gamma}\log^{q}N_{\Gamma})$ is almost linear in $N_{\Gamma}$ -- 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 $\mathcal{O}(N_{\Gamma}\log^{q}N_{\Gamma})$, while in the general case the FEM discretisation leads to the complexity $O(N_{\Omega}\log^{q}N_{\Omega})$, where $N_{\Omega} $ is the number of degrees of freedom in the domain.
