Plane Implicit Multigrid Schemes for Anisotropic Elliptic Equations

N. Duane Melson

Mail Stop 128
NASA, Langley Research Center
Hampton, VA 23681


We discuss the behavior of traditional plane relaxation methods as multigrid smoothers for the solution of a discrete anisotropic elliptic model problem on cell-centered grids. Based on numerical experiments and local mode analysis, we compare the different methods considering their smoothing factor for strong anisotropies with Dirichlet boundary conditions. There are important discrepancies between periodic and Dirichlet boundary conditions for practical grid sizes that significantly change the multigrid smoothing factor of some plane relaxation methods for strong anisotropies. For Dirichlet boundary conditions, the performance deteriorates as the number of cells per side in the grid increases, tending to the performance obtained with periodic boundary conditions. However, due to computer memory limitations, the number of cells per side is usually much lower in three-dimensional cases than in one or two-dimensional cases.

An alternating direction plane Gauss-Seidel method with lexicographic ordering gives very good convergence rates for very strong anisotropies. The zebra Gauss-Seidel method does not perform as well on the cell-centered grids in the present work as the lexicographically ordered method but the four-color Gauss-Seidel method is found have the best numerical and architectural properties of the methods considered. Although alternating direction plane relaxation is simpler and more robust than other robust approaches, it is not used in industrial and production codes because it requires the solution of a two-dimensional problem for each plane in each direction. We show that an exact solution of each plane is not necessary and that a single 2-D multigrid cycle gives the same result as an exact solution, in much less execution time. Parallelization of the two-dimensional multigrid cycles and the kernel of the three-dimensional implicit solver is also discussed. As a result, alternating-plane smoothers are found to be highly efficient smoothers for the anisotropic elliptic discrete operator.

In many applications, domain decomposition (or multiblock grids) are used to increase geometric flexibility. This inherently limits the range of an implicit smoother to only the portion of the computational domain in the current block. Previous analysis suggests that this can have a detrimental effect on the performance of plane implicit schemes. This effect will be studied numerically in the full paper. A multiblock stategy does open up the possibility of using different smoothers for different portions of the domain where the choice of smoother is based on a minimization of operation count while retaining optimum smoothing performance. An example of this would be using a plane implicit smoother in the portions of the domain that have strong anisotropies while using a point smoother in the regions that are isotropic. The use of various smoothers for different portions of the grid will also be explored in the full paper.

N. Duane Melson
Ignacio M. Llorente
James L. Thomas