Coarse grid approximation governed by local

Abstract

Solving discrete boundary value problems with the help of an appropriate multigrid method [1, 3, 4, 5] necessitates the construction of a sequence of coarse grids with corresponding coarse grid approximations for the given fine grid discretization. Popular choices in this context are the Galerkin coarse grid approximation (GCA) and the use of the same discretization on the coarser grids as on the fine grid with properly adjusted mesh sizes (DCA). In this paper we propose an alternative strategy to select the required coarse grid discretizations within a multigrid solution method. It can be applied to discretization operators that can be locally represented by a stencil. The coarse grid approximations are constructed by minimizing a certain low-frequency L ^ 2-norm. More precisely, a coarse grid discretization is chosen in such a way, that its Fourier symbol is a best approximation (w.r.t. low frequencies) of the Fourier symbol of the fine grid operator. This strategy is abbreviated by FCA since the design of coarse grid approximations is based on local Fourier analysis~[1, 4, 5, 6]. The entries of the coarse grid stencils are simply given by linear combinations of the fine grid entries. As a consequence, FCA can be considered as a black-box method to construct coarse grid operators. This method has been successfully applied to (anisotropic) diffusion equations, operators with mixed derivatives, problems with dominant convection, and operators involving jumping coefficients. For nicely elliptic examples, FCA resembles the DCA approach, whereas for more difficult applications (w.r.t. an efficient multigrid treatment) it behaves similarly to GCA based on operator-dependent transfers [2,7].