Matrix-dependent prolongations and restrictions in a blackbox multigrid solver

Abstract

Multigrid methods are studied for the solution of linear systems resulting from the 9-point discretization of a general linear second-order elliptic partial differential equation in two dimensions. The rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous, or when dominating first-order terms are present. These difficulties may be overcome by choosing the prolongation and restriction operators in a special way. A novel way to do this is proposed. As a result, a blackbox solver (written in standard FORTRAN 77) has been developed.

Numerical experiments for several hard test problems are described and comparison is made with other algorithms: the standard MG method and a method introduced by Kettler. A significant improvement of robustness and efficiency is found.