Black box multigrid with coarsening by a factor of three

Abstract

Black Box Multigrid (BoxMG) is a robust variational multigrid solver for diffusion equations on logically structured grids. BoxMG standardly uses coarsening by a factor of two. It handles cell-centered discretizations on logically rectangular grids by treating the cell-centers as the unknowns to be coarsened. Such a strategy does not preserve the cell structure. That is, coarse-grid cells are not the union of fine-grid cells. In some applications, such as local grid refinement, it is desirable that the cell structure be preserved. In this paper, we develop a method that employs coarsening by a factor of three. It is a natural generalization of standard BoxMG, using operator-induced interpolation (which approximately preserves the continuity of the normal flux), restriction as the transpose of interpolation, and Galerkin coarsening. In addition, we introduce a new colored block Gauss-Seidel scheme that is motivated by the form of the interpolation operator, dubbed 'pattern' relaxation. We present numerical results that demonstrate robustness of this method with respect to discontinuous diffusion coefficients, boundary conditions, and grid dimension. © 2010 by John Wiley & Sons, Ltd.