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Existence of $ \mathcal{H}$-matrix approximants to the inverses of BEM matrices: The simple-layer operator


Authors: Markus Faustmann, Jens Markus Melenk and Dirk Praetorius
Journal: Math. Comp. 85 (2016), 119-152
MSC (2010): Primary 65F05; Secondary 65N38, 65F30, 65F50
Published electronically: June 18, 2015
MathSciNet review: 3404445
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Abstract: We consider the question of approximating the inverse $ \mathbf W = \mathbf V^{-1}$ of the Galerkin stiffness matrix $ \mathbf V$ obtained by discretizing the simple-layer operator $ V$ with piecewise constant functions. The block partitioning of $ \mathbf W$ is assumed to satisfy any one of several standard admissibility criteria that are employed in connection with clustering algorithms to approximate the discrete BEM operator $ \mathbf V$. We show that $ \mathbf W$ can be approximated by blockwise low-rank matrices such that the error decays exponentially in the block rank employed. Similar exponential approximability results are shown for the Cholesky factorization of $ \mathbf V$.


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Additional Information

Markus Faustmann
Affiliation: Institute for Analysis and Scientific Computing (Inst. E 101), Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Wien, Austria
Email: markus.faustmann@tuwien.ac.at

Jens Markus Melenk
Affiliation: Institute for Analysis and Scientific Computing (Inst. E 101), Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Wien, Austria
Email: melenk@tuwien.ac.at

Dirk Praetorius
Affiliation: Institute for Analysis and Scientific Computing (Inst. E 101), Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Wien, Austria
Email: dirk.praetorius@tuwien.ac.at

DOI: https://doi.org/10.1090/mcom/2990
Received by editor(s): November 20, 2013
Received by editor(s) in revised form: July 28, 2014, and August 11, 2014
Published electronically: June 18, 2015
Dedicated: Dedicated to Wolfgang Hackbusch on the occasion of his 65th birthday
Article copyright: © Copyright 2015 American Mathematical Society