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Robust norm equivalencies for diffusion problems

Authors: Michael Griebel, Karl Scherer and Marc Alexander Schweitzer
Journal: Math. Comp. 76 (2007), 1141-1161
MSC (2000): Primary 65N55, 65F35; Secondary 65N30, 65F10
Published electronically: February 7, 2007
MathSciNet review: 2299769
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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

Karl Scherer
Affiliation: Institut für Angewandte Mathematik, Universität Bonn, Germany

Marc Alexander Schweitzer
Affiliation: Institut für Numerische Simulation, Universität Bonn, Germany

Keywords: Norm equivalency, multilevel method, preconditioning, robustness
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.
Article copyright: © Copyright 2007 American Mathematical Society