An Evaluation of Parallel Multigrid as a Solver and a Preconditioner for Singularly Perturbed Problems

Abstract

In this paper we try to achieve h-independent convergence with preconditioned GMRES ([Y. Saad and M. H. Schultz, SIAM J. Sci. Comput., 7 (1986), pp. 856--869]) and BiCGSTAB ([H. A. Van der Vorst, SIAM J. Sci. Comput., 13 (1992), pp. 63--644]) for two-dimensional (2D) singularly perturbed equations. Three recently developed multigrid methods are adopted as a preconditioner. They are also used as solution methods in order to compare the performance of the methods as solvers and as preconditioners. Two of the multigrid methods differ only in the transfer operators. One uses standard matrix-dependent prolongation operators from [J. E. Dendy Jr., J. Comput. Phys., 48 (1982), pp. 366--386], [W. Hackbusch, Multi-grid Methods and Applications, Springer, Berlin, 1985]. The second uses "upwind" prolongation operators, developed in [P. M. de Zeeuw, J. Comput.\ Appl.\ Math., 33 (1990), pp. 1--27]. Both employ the Galerkin coarse grid approximation and an alternating zebra line Gauss--Seidel smoother. The third method is based on the block LU decomposition of a matrix and on an approximate Schur complement. This multigrid variant is presented in [A. Reusken, A Multigrid Method Based on Incomplete Gaussian Elimination, University of Eindhoven, the Netherlands, 1995]. All three multigrid algorithms are algebraic methods. The eigenvalue spectra of the three multigrid iteration matrices are analyzed for the equations solved in order to understand the convergence of the three algorithms. Furthermore, the construction of the search directions for the multigrid preconditioned GMRES solvers is investigated by the calculation and solution of the minimal residual polynomials. For Poisson and convection-diffusion problems all solution methods are investigated and evaluated for finite volume discretizations on fine grids. The methods have been parallelized with a grid partitioning technique and are compared on an MIMD machine.