Sparse Multiple Semicoarsened Grids

Robert D. Falgout

Center for Applied Scientific Computing
Lawrence Livermore National Laboratory
P.O. Box 808, L-561
Livermore, CA 94551


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 [4]. The robustness of MSG is studied in [5], and a new, more efficient, variant is proposed in [6]. Our approach for improving efficiency is based on the relationship between the MSG algorithm, the hierarchical-basis multigrid method [10], 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 no. W-7405-Eng-48.

[1] S. Schaffer, A semi-coarsening multigrid method for elliptic partial differential equations with highly discontinuous and anisotropic coefficients, SIAM J. Sci. Comput., 20 (1998), pp. 228-242.
[2] P. N. Brown, R. D. Falgout, and J. E. Jones, Semicoarsening multigrid on distributed memory machines. To appear in the SIAM Journal on Scientific Computing special issue on the Fifth Copper Mountain Conference on Iterative Methods. Also available as LLNL technical report UCRL-JC-130720, 1999.
[3] W. A. Mulder, A new multigrid approach to convection problems, J. Comput. Phys., 83 (1989), pp. 303--323.
[4] N. H. Naik and J. R. Rosendale, The improved robustness of multigrid elliptic solvers based on multiple semicoarsened grids, SIAM J. Numer. Anal., 30 (1993), pp. 215-229.
[5] C. W. Oosterlee and P. Wesseling, On the robustness of a multiple semi-coarsened grid method, Z. Angew. Math. Mech., 75 (1995), pp. 251-257.
[6] T. Washio and C. W. Oosterlee, Flexible multiple semicoarsening for three-dimensional singularly perturbed problems, SIAM J. Sci. Comput., 19 (1998), pp. 1646-1666.
[7] W. Hackbusch, The frequency decomposition multi-grid method, in Multigrid Methods IV, Proceedings of the Fourth European Multigrid Conference, Amsterdam, July 6-9, 1993, vol. 116 of ISNM, Basel, 1994, Birkhauser, pp. 43-56.
[8] J. E. Dendy and C. C. Tazartes, Grandchild of the frequency decomposition multigrid method, SIAM J. Sci. Comput., 16 (1995), pp. 307--319.
[9] J. E. Dendy, Revenge of the semicoarsening frequency decomposition multigrid method, SIAM J. Sci. Comput., 18 (1997), pp. 430-440.
[10] R. E. Bank, T. Dupont, and H. Yserentant, The hierarchical basis multigrid method, Numer. Math., 52 (1988), pp. 427-458.
[11] C. Zenger, Sparse grids, in Parallel algorithms for partial differential equations: Proceedings of the Sixth GAMM-Seminar, Kiel, Jan. 1990, vol. Notes on Numerical Fluid Mechanics, vol. 31, Vieweg, Braunschweig, 1991.
[12] M. Griebel, M. Schneider, and C. Zenger, A combination technique for the solution of sparse grid problems, in Proceedings of the IMACS International Symposium on Iterative Methods in Linear Algebra, Amsterdam, 1992, Elsevier, pp. 263-281.