Matrix-Based Multigrid for Locally Refined Meshes
In this chapter, we describe a matrix-based multigrid method suitable for complicated nonuniform meshes obtained from local refinement. Under some algebraic assumptions such as diagonal dominance of the coefficient matrices, we derive an aposteriori upper bound for the condition number of the V(0,0)-cycle. This result applies also to diffusion problems with variable and even discontinuous coefficients, even when the discontinuity lines don’t align with the coarse mesh. Furthermore, the upper bound is independent of the meshsize and the jump in the diffusion coefficient. Of course, the actual application of the multigrid method uses more efficient cycles such as the V(1,1)-cycle. Still, the theoretical result indicates that the nearly singular eigenvectors of A are indeed well approximated on the coarse grids, so the remaining error modes can be well handled by the relaxation.
KeywordsCondition Number Coarse Grid Coarse Mesh Multigrid Method Discontinuity Line
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