Abstract. This paper contains the first published numerical results and analysis of the behavior of alternating plane relaxation methods as multigrid smoothers for cell-centered grids. The results are very satisfactory: plane smoothers work well in general and their performance improves considerably for strong anisotropies in the right direction because they effectively become exact solvers. In fact, the convergence rate decreases (improves) linearly with increasing anisotropy strength. The methods compared are plane Jacobi with damping, plane Jacobi with partial damping, plane Gauss-Seidel, plane zebra Gauss-Seidel, and line Gauss-Seidel. Based on numerical experiments and local mode analysis, the smoothing factor and cost per cycle of the different methods in the presence of strong anisotropies for Dirichlet boundary conditions are compared. A four-color Gauss-Seidel method is found to have the best numerical and architectural properties of the methods considered in the present work. Although alternating direction plane relaxation schemes are simpler and more robust than other approaches, they are not currently used in industrial and production codes because they require the solution of a two-dimensional problem for each plane in each direction. We verify the theoretical predictions of Thole and Trottenberg that an exact solution of each plane is not necessary; in fact, a single two-dimensional multigrid cycle gives the same result as an exact solution, in much less execution time. As a result, alternating-plane smoothers are found to be highly efficient multigrid smoothers for anisotropic elliptic problems.

The aim of this paper is to present an easy and efficient method to implement alternating-line processes on current parallel computers. First we show how data locality has an important impact on global efficiency, which leads us to the conclusion that one-dimensional decompositions are the most convenient ones for 2D problems. Once this is asserted, a parallel algorithm is presented for the solution of the distributed tridiagonal systems along the partitioned direction. The key idea is to pipeline the simultaneous resolution of many systems of equations, not parallelising each resolution separately. This approach presents good numerical and architectural properties, in terms of memory usage and data locality, and high parallel efficiencies are obtained. For the case of alternating-line processes, the election of the optimal decomposition is studied. The experimental results have been obtained on a Cray T3E. 1. Introduction Alternating-line processes are widely applied in scientific co...

Standard multigrid methods are not well suited for problems with anisotropic discrete operators, which can occur, for example, on grids that are stretched in order to resolve a boundary layer. One of the most e#cient approaches to yield robust methods is the combination of standard coarsening with alternating-direction plane relaxation in the three dimensions. However, this approach may be di#cult to implement in codes with multiblock structured grids because there may be no natural definition of global lines or planes. This inherent obstacle limits the range of an implicit smoother to only the portion of the computational domain in the current block. This report studies in detail, both numerically and analytically, the behavior of blockwise plane smoothers in order to provide guidance to engineers who use block-structured grids. The results obtained so far show alternating-direction plane smoothers to be very robust, even on multiblock grids. In common computational fluid dynamics multiblock simulations, where the number of subdomains crossed by the line of a strong anisotropy is low (up to four), textbook multigrid convergence rates can be obtained with a small overlap of cells between neighboring blocks.

Abstract. Standard multigrid methods are not well suited for problems with anisotropic discrete operators, which can occur, for example, on grids that are stretched in order to resolve a boundary layer. One of the most efficient approaches to yield robust methods is the combination of standard coarsening with alternating-direction plane relaxation in the three dimensions. However, this approach may be difficult to implement in codes with multiblock structured grids because there may be no natural definition of global lines or planes. This inherent obstacle limits the range of an implicit smoother to only the portion of the computational domain in the current block. This report studies in detail, both numerically and analytically, the behavior of blockwise plane smoothers in order to provide guidance to engineers who use block-structured grids. The results obtained so far show alternating-direction plane smoothers to be very robust, even on multiblock grids. In common computational fluid dynamics multiblock simulations, where the number of subdomains crossed by the line of a strong anisotropy is low (up to four), textbook multigrid convergence rates can be obtained with a small overlap of cells between neighboring blocks. Key words. robust multigrid methods, multiblock grids, anisotropic discrete operators Subject classification. Applied Mathematics 1. Introduction and previous work. Standard multigrid techniques are efficient methods for solving

In this article we introduce an asymptotic preserving scheme designed to compute the solution of a two dimensional elliptic equation presenting large anisotropies. We focus on an anisotropy aligned with one direction, the dominant part of the elliptic operator being supplemented with Neumann boundary conditions. A new scheme is introduced which allows an accurate resolution of this elliptic equation for an arbitrary anisotropy ratio. 1