A highly parallel multigrid-like method for the solution of the Euler equations

We consider a highly parallel multigrid-like method for the solution of the two dimensional steady Euler equations. The new method, introduced as filtering multigrid, is similar to a standard multigrid scheme in that convergence on the finest grid is accelerated by iterations on coarser grids. In the filtering method, however, additional fine grid subproblems are processed concurrently with coarse grid computations to further accelerate convergence. These additional problems are obtained by splitting the residual into a smooth and an oscillatory component. The smooth component is then used to form a coarse grid problem (similar to standard multigrid) while the oscillatory component is used for a fine grid subproblem. The primary advantage in the filtering approach is that fewer iterations are required and that most of the additional work per iteration can be performed in parallel with the standard coarse grid computations. We generalize the filtering algorithm to a version suitable for nonlinear problems. We emphasize that this generalization is conceptually straight-forward and relatively easy to implement. In particular, no explicit linearization (e.g., formation of Jacobians) needs to be performed (similar to the FAS multigrid approach). We illustrate the nonlinear version by applying it to the Euler equations, and presenting numerical results. Finally, a performance evaluation is made based on execution time models and convergence information obtained from numerical experiments

Analysis of a parallel multigrid algorithm

The parallel multigrid algorithm of Frederickson and McBryan (1987) is considered. This algorithm uses multiple coarse-grid problems (instead of one problem) in the hope of accelerating convergence and is found to have a close relationship to traditional multigrid methods. Specifically, the parallel coarse-grid correction operator is identical to a traditional multigrid coarse-grid correction operator, except that the mixing of high and low frequencies caused by aliasing error is removed. Appropriate relaxation operators can be chosen to take advantage of this property. Comparisons between the standard multigrid and the new method are made

Auxiliary Space Preconditioners for Mixed Finite Element Methods

Summary. This paper is devoted to study of an auxiliary spaces preconditioner for H(div) systems and its application in the mixed formulation of second order elliptic equations. Extensive numerical results show the efficiency and robustness of the algorithms, even in the presence of large coefficient variations. For the mixed formulation of elliptic equations, we use the augmented Lagrange technique to convert the solution of the saddle point problem into the solution of a nearly singular H(div) system. Numerical experiments also justify the robustness and efficiency of this scheme

Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines

Algebraic multigrid methods offer the hope that multigrid convergence can be achieved (for at least some important applications) without a great deal of effort from engineers and scientists wishing to solve linear systems. In this paper the authors consider parallelization of the smoothed aggregation multi-grid method. Smoothed aggregation is one of the most promising algebraic multigrid methods. Therefore, developing parallel variants with both good convergence and efficiency properties is of great importance. However, parallelization is nontrivial due to the somewhat sequential aggregation (or grid coarsening) phase. In this paper, they discuss three different parallel aggregation algorithms and illustrate the advantages and disadvantages of each variant in terms of parallelism and convergence. Numerical results will be shown on the Intel Teraflop computer for some large problems coming from nontrivial codes: quasi-static electric potential simulation and a fluid flow calculation

Parallel Smoothed Aggregation Multigrid: Aggregation Strategies on Massively Parallel Machines

Algebraic multigrid methods o#er the hope that multigrid con vergen ce can be achieved (for at least some importan t application s) without a great deal of e#ort from en gin eers an d scien tists wishin g to solvelin ear systems. In this paper we con sider parallelization of the smoothed aggregation multigrid method. Smoothed aggregation is on e of the most promisin g algebraic multigrid methods. Therefore, developin g parallel varian ts with both good con vergen ce an de#cien cy properties is of great importan ce. However, parallelization isn on trivial due to the somewhat sequen tial aggregation (or grid coarsen in g) phase. In this paper, we discuss three di#eren t parallel aggregation algorithms an d illustrate the advan tages an d disadvan tages of each varian t in terms of parallelism an d con vergen ce. Numerical results will be shown on the In tel Teraflop computer for some large problems comin g from n on trivial codes: quasi-static electric poten tial simulation an d a fluid..

SURVEY OF PARALLEL MULTIGRID ALGORITHMS.

The multigrid algorithm is a fast and efficient (in fact provably optimal) method for solving a wide class of integral and partial differential equations. In addition, it is a natural choice for implementation on parallel computers because of the parallelism inherent in the algorithm. Over the past several years, there has been increasing research into parallel multigrid algorithms, ranging from purely theoretical studies to actual codes running on real parallel computers, some built with multigrid algorithms in mind. It is our goal in this paper to provide a brief but structured account of this field of research