Algebraic Multigrid Solver for Discrete Elliptic Second Order Problems

Ferdinand Kickinger

April 17, 1996

Abstract

In this paper, an Algebraic Multigrid Solver (AMG++) is presented. This solver is especially suited for the fast solution of large sparse systems A_{h}u_{h} = f_{h} of algebraic equations arrizing from finite element (FE) discretisation of second-order elliptic boundary value problems (BVP) on unstructured refined meshes in one, two and three dimensions as well. The only information used is recovered from the stiffness matrix A_{h}. More precisely, the information hidden in the stiffness matrix allows us, to construct ”coarse grids”, the grid intergrid transfer operators (restriction and prolongation) and the coarse grid matrices. These tools together with some smoother and a solver for the equations on the coarsest grid build up a complete multigrid algorithm. In special cases, we observe, that the AMS completely coincides with some standard geometrical multigrid algorithms. The AMS has been applied to two and three dimensional examples. The algebraic multigrid algorithm shows the same numerical behaviour as it is known from the geometrical multigrid algorithms. Symmetric variants of the algebraic multigrid method can be used for preconditioning a conjugate gradient solver. This combination is more robust and improves the convergence rate considerably.