An automatic multigrid method for the solution of sparse linear systems

Yair Shapira

Mathematics Department, Technion, Haifa 32000, Israel


For the solution of linear systems which arise from elliptic PDE's, automatic version of the Multigrid method is presented. This version relies on the algebraic system solely, and not on the original PDE. Numerical experiments show that for the Poisson equation its rate of convergence is close to that of the classical Multigrid method. Nevertheless, it is robust in the sense that its fast convergence is conserved in other classes of problems: non-symmetric, hyperbolic (even with closed characteristics) and problems on non-uniform grids. When applied with a Krylov-type acceleration, it copes with anisotropic, discontinues and indefinite problems as well. No special treatment of sub-domains (e.g., boundaries) is needed.