Towards Algebraic Multigrid

for Elliptic Problems of Second Order

Dietrich Braess, Bochum

Abstract

An algebraic multigrid method is developed which can be used as a preconditioner for the solution of linear systems of equations with positive definite matrices. The method is directed to equations which arise from the discretization of elliptic equations of second order, but only the matrix is the source for the information used by the algorithm. One has only to know whether the matrix stems from a 2-dimensional or 3-dimensional problem and whether the elliptic equations are scalar equations or belong to a system.

A cycle with the AMG preconditioner costs about 2.5 as much as a cycle of a cg-iteration with a standard preconditioner. The initialization is as expensive as 3 cycles.

Numerical examples show that a convergence rate of 0.13 is achieved for the Poisson equation on a rectangular grid. When problems of planar elasticity are treated, the number of iterations is nearly twice the number required for the Poisson equation. Examples with irregular meshes have also successfully been treated.