Much work has been done developing robust multigrid methods for solving diffusion equations on structured grids, especially when discontinuous coefficients and anisotropy are present (see e.g., [1-9]). However, all of these methods have relatively high computational costs and memory costs. The goal of this work is to develop an algorithm that is both robust and efficient.
We will present a method based on a modification of the multiple semicoarsened grids (MSG) method in . The robustness of MSG is studied in , and a new, more efficient, variant is proposed in . Our approach for improving efficiency is based on the relationship between the MSG algorithm, the hierarchical-basis multigrid method , and the sparse grids method [11,12].
The sparse grids method is actually a discretization method that considerably reduces computational and storage costs with little loss in discretization accuracy. The method uses hierarchical-basis functions defined on a subset of the MSG grids, and uses the so-called combination technique to implicitly define the discretization on the sparse grid. It is important to note that although the sparse grids method and the MSG method both provide solutions to the diffusion problem of interest, they do not solve them on the same discrete grid. We consider solving the diffusion equation with the MSG method, but with only a subset of the MSG grids. If we use the same grids as in the sparse grids method, one can think of this approach as using the sparse grids discretization to form the coarse grid problem in a two-level multigrid method. Since the sparse grids discretization is nearly as accurate as the standard Cartesian discretization, we expect this method to work well.
This work was performed under the auspices of the U.S. Department of
Energy by Lawrence Livermore National Laboratory under contract
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