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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 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.

Received:
Feb 28, 2002
Published:
Feb 28, 2002
MSC Codes:
35C20, 65F05, 65F50, 65N30
Keywords:
hierarchical matrices, inverse fe-matrix, jumping coefficients, green's function

Related publications

inJournal
2003 Repository Open Access
Mario Bebendorf and Wolfgang Hackbusch

Existence of \(\mathscr {H}\)-matrix approximants to the inverse FE-matrix of elliptic operators with \(L^\infty\)-coefficients

In: Numerische Mathematik, 95 (2003) 1, pp. 1-28