We have decided to discontinue the publication of preprints on our preprint server end of 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
21/2002
Existence of $\mathcal{H}$-Matrix Approximants to the Inverse FE-Matrix of Elliptic Operators with $L^{\infty}$-Coefficients
Mario Bebendorf and Wolfgang Hackbusch
Abstract
This article deals with the existence of blockwise low-rank approximants --- so-called $\mathcal{H}$-matrices --- to inverses of FEM matrices in the case of uniformly elliptic operators with $L^{\infty}$-coefficients. Unlike operators arising from boundary element methods for which the $\mathcal{H} $-matrix theory has been extensively developed, the inverses of these operators do not benefit from the smoothness of the kernel function. However, it will be shown that the corresponding Green functions can be approximated by degenerate functions giving rise to the existence of blockwise low-rank approximants of FEM inverses. Numerical examples confirm the correctness of our estimates. As a side-product we analyse the $\mathcal{H}$-matrix property of the inverse of the FE mass matrix.
