AN EVALUATION OF PARALLEL MULTIGRID AS A SOLVER AND A PRECONDITIONER FOR SINGULARLY PERTURBED PROBLEMS

In this paper we try to achieve h-independent convergence with precond itioned GMRES ([Y. Saad and M. H. Schultz, SIAM J. Sci. Comput., 7 (19 86), pp. 856-869]) and BiCGSTAB ([H. A. Van der Vorst, SIAM J. Sci. Co mput., 13 (1992), pp. 63-644]) for two-dimensional (2D) singularly per turbed equations. Three recently developed multigrid methods are adopt ed as a preconditioner. They are also used as solution methods in orde r to compare the performance of the methods as solvers and as precondi tioners. Two of the multigrid methods differ only in the transfer oper ators. One uses standard matrix-dependent prolongation operators from [J. E. Dendy Jr., J. Comput. Phys., 48 (1982), pp. 366-386], [W. Hackb usch, Multi-grid Methods and Applications, Springer, Berlin, 1985]. Th e second uses ''upwind'' prolongation operators, developed in [P. M. d e Zeeuw, J. Comput. Appl. Math., 33 (1990), pp. 1-27]. Both employ the Galerkin coarse grid approximation and an alternating zebra line Gaus s-Seidel smoother. The third method is based on the block LU decomposi tion of a matrix and on an approximate Schur complement. This multigri d variant is presented in [A. Reusken, A Multigrid Method Based on Inc omplete Gaussian Elimination, University of Eindhoven, the Netherlands , 1995]. All three multigrid algorithms are algebraic methods. The eig envalue spectra of the three multigrid iteration matrices are analyzed for the equations solved in order to understand the convergence of th e three algorithms. Furthermore, the construction of the search direct ions 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 in vestigated and evaluated for finite volume discretizations on fine gri ds. The methods have been parallelized with a grid partitioning techni que and are compared on an MIMD machine.