Approximation of solution operators of elliptic partial differential equations by - and ²-matrices

Steffen Börm

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Submission date: 06. Sep. 2007 (revised version: September 2007)
Pages: 27
published in: Numerische Mathematik, 115 (2010) 2, p. 165-193 
DOI number (of the published article): 10.1007/s00211-009-0278-7
Bibtex
MSC-Numbers: 65N22, 65N30, 65F05
Keywords and phrases: Hierarchical matrix, H^2-matrix, PDE
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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 - and -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 - and -matrix approximations of the entire matrices.