Matrix-dependent Prolongations and Restrictions in a Black-box Multigrid Solver

Multigrid methods are studied for the solution of linear systems resulting from the 9-point discretization of a general linear second-order elliptic partial di erential equation in two dimensions. The rate of convergence of standard multigrid methods often deteriorates when the coe cients in the di erential equation are discontinuous, or when dominating rst-order terms are present. These di culties 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 black-box 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 signi cant improvement of robustness and e ciency is found. Note: This chapter has been published in J. Comput. Appl. Math. 33 (1990) 1{27. 3.