266 IGPM266.pdf        December 2006
TITLE On the Robustness of a Multigrid Method for Anisotropic Reaction-Diffusion Problems
AUTHORS Arnold Reusken, Marcus Soemers
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 [5]. 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.
KEYWORDS anisotropic reaction-diffusion problem, robust multigrid method
DOI 10.1007/s00607-007-0232-4
80(4), 299-317 (2007)