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MiS Preprint
85/2007

Approximation of solution operators of elliptic partial differential equations by ${\mathcal H}$- and ${\mathcal H}^2$-matrices

Steffen Börm

Abstract

We investigate the problem of computing the inverses of stiffness matrices resulting from the finite element discretization of elliptic partial differential equations. Since the solution operators are non-local, the inverse matrices will in general be dense, therefore they cannot be represented by standard techniques. In this paper, we prove that these matrices can be approximated by ${\mathcal H}$- and ${\mathcal H}^2$-matrices. The key results are existence proofs for local low-rank approximations of the solution operator and its discrete counterpart, which give rise to error estimates for ${\mathcal H}$- and ${\mathcal H}^2$-matrix approximations of the entire matrices.

Received:
Sep 6, 2007
Published:
Sep 6, 2007
MSC Codes:
65N22, 65N30, 65F05
Keywords:
Hierarchical matrix, H^2-matrix, PDE

Related publications

inJournal
2010 Repository Open Access
Steffen Börm

Approximation of solution operators of elliptic partial differential equations by \(\mathscr {H}\)- and \(\mathscr {H}^2\)-matrices

In: Numerische Mathematik, 115 (2010) 2, pp. 165-193