A General Approach to Algebraic Multigrid Methods

In this report a general approach to algebraic multigrid methods for problems arising from the nite element discretization of a second order, self-adjoint, elliptic partial diierential equation is proposed. Special attention is paid to the coarsening process and the transfer operators. In order to construct a more exible method an auxiliary matrix is introduced which represents a virtual nite element mesh. In addition this auxiliary matrix is related to the degrees of freedom of the system matrix. The coarsening is performed on the auxiliary matrix, and after deening appropriate transfer operators for the system and the auxiliary matrix, a coarse system can be constructed by Galerkin's method. Moreover, a necessary condition imposed on the corresponding transfer operators is given such that the properties of the ne level system hands over to a coarse level system. It turns out that this approach is a generalization of many existing algebraic multigrid methods. Numerical examples are given which show the eeciency and exibility of the proposed method.