Multigrid methods for parabolic problems with local mesh refinement

Chisup Kim

Department of Mathematics
Duke University
Durham, NC 27708-0320


Multigrid methods for parablolic problems have been analyzed in [2] when the grid is (quasi) uniform. In the current work, we consider multigrid methods for parabolic problems on a locally refined grid. On each level, smoothing is done only on subspaces corresponding to the refined regions. Such a method is useful, for example, when modeling the electrical activity in human heart where strong electrical shocks are present [3]. Uniform convergence of multigrid methods is established independently of the mesh size and time step size in a certain time step dependent norm by applying the results in [1].


[1] J. H. Bramble and J. E. Pasciak, New estimates for multigrid algorithms including the V-cycle, Math. Comp., 60 (1993), pp. 447--471.
[2] S. Larsson, V. Thom\'{e}e, and S. Z. Zhou, On multigrid methods for parabolic problems, J. Comput. Math., 13 (1995), pp. 193--205.
[3] J. A. Trangenstein and K.~Skoubine, Operator splitting and adaptive mesh refinement for the Fitzhugh-Nagumo problem, preprint, 2001.