On the robustness of a multigrid method for anisotropic reaction-diffusion problems

Authors:
A. Reusken;M. Soemers
Affiliations:
RWTH-Aachen, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52056, Aachen, Germany;RWTH-Aachen, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52056, Aachen, Germany
Venue:
Computing
Year:
2007

Citing 5
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Abstract

In this paper, we consider a reaction-diffusion boundary value problem in a three-dimensional thin domain. The very different length scales in the geometry result in an anisotropy effect. Our study is motivated by a parabolic heat conduction problem in a thin foil leading to such anisotropic reaction-diffusion problems in each time step of an implicit time integration method [7]. The reaction-diffusion problem contains two important parameters, namely ε 0 which parameterizes the thickness of the domain and μ 0 denoting the measure for the size of the reaction term relative to that of the diffusion term. In this paper we analyze the convergence of a multigrid method with a robust (line) smoother. Both, for the W- and the V-cycle method we derive contraction number bounds smaller than one uniform with respect to the mesh size and the parameters ε and μ.