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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Robust norm equivalencies for diffusion problems
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by Michael Griebel, Karl Scherer and Marc Alexander Schweitzer PDF
Math. Comp. 76 (2007), 1141-1161 Request permission

Abstract:

Additive multilevel methods offer an efficient way for the fast solution of large sparse linear systems which arise from a finite element discretization of an elliptic boundary value problem. These solution methods are based on multilevel norm equivalencies for the associated bilinear form using a suitable subspace decomposition. To obtain a robust iterative scheme, it is crucial that the constants in the norm equivalence do not depend or depend only weakly on the ellipticity constants of the problem. In this paper we present such a robust norm equivalence for the model problem $- \nabla \omega \nabla u=f$ with a scalar diffusion coefficient $\omega$ in $\Omega \subset \mathbb {R}^2$. Our estimates involve only very weak information about $\omega$, and the results are applicable for a large class of diffusion coefficients. Namely, we require $\omega$ to be in the Muckenhoupt class $A_{1}(\Omega )$, a function class well-studied in harmonic analysis. The presented multilevel norm equivalencies are a main step towards the realization of an optimal and robust multilevel preconditioner for scalar diffusion problems.
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Additional Information
  • Michael Griebel
  • Affiliation: Institut für Numerische Simulation, Universität Bonn, Germany
  • MR Author ID: 270664
  • Email: griebel@ins.uni-bonn.de
  • Karl Scherer
  • Affiliation: Institut für Angewandte Mathematik, Universität Bonn, Germany
  • Email: scherer@iam.uni-bonn.de
  • Marc Alexander Schweitzer
  • Affiliation: Institut für Numerische Simulation, Universität Bonn, Germany
  • Email: m.a.schweitzer@ins.uni-bonn.de
  • Received by editor(s): August 4, 2004
  • Received by editor(s) in revised form: August 3, 2006
  • Published electronically: February 7, 2007
  • Additional Notes: The authors were supported in part by the Sonderforschungsbereich 611 Singuläre Phänomene und Skalierung in Mathematischen Modellen sponsored by the Deutsche Forschungsgemeinschaft.
  • © Copyright 2007 American Mathematical Society
  • Journal: Math. Comp. 76 (2007), 1141-1161
  • MSC (2000): Primary 65N55, 65F35; Secondary 65N30, 65F10
  • DOI: https://doi.org/10.1090/S0025-5718-07-01973-4
  • MathSciNet review: 2299769