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A New Proof for Existence of H-Matrix Approximants to the Inverse of FEM Matrices: The Dirichlet Problem for the Laplacian

  • Markus Faustmann
  • Jens M. Melenk
  • Dirk Praetorius
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 95)

Abstract

We study the question of approximability of the inverse of the FEM stiffness matrix for the Laplace problem with Dirichlet boundary conditions by blockwise low rank matrices such as those given by the \(\mathcal{H}\)-matrix format introduced in Hackbusch (Introd \(\mathcal{H}\)-Matrices Comput 62(2):89–108, 1999). We show that exponential convergence in the local block rank r can be achieved. Unlike prior works Bebendorf and Hackbusch (Numer Math 95(1):1–28, 2003) and Börm (Numer Math 115(2):165–193, 2010) our analysis avoids any a priori coupling \(r = \mathcal{O}({\vert \log }^{\alpha }h\vert )\) of r and the mesh width h. Moreover, the techniques developed can be used to analyze other boundary conditions as well.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Markus Faustmann
    • 1
  • Jens M. Melenk
    • 1
  • Dirk Praetorius
    • 1
  1. 1.Institut für Analysis und Scientific ComputingTechnische Universität WienWienAustria

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