Integration of the Composite Mesh Technique with the Multigrid Method

Abstract

The Composite Finite Element Mesh method is useful for discretization error estimation and, in addition, for solution improvement with no appreciable increment in the computational cost. The technique consists in redefine over a given mesh the linear operator that arises from the discretization of a partial differential equation. This operator is modified according to an appropriate linear combination between the operators of the given mesh and of a coarse mesh, which must be a coarsening of the first one. On the other hand, Multigrid methods solve a linear system using systems of several sizes resulting from different discretization levels. This feature motivates the study of the application of the Multigrid strategy in conjunction with the Composite Mesh technique. In this work, we propose a scheme for solving the linear system arising from the Composite Mesh strategy using a Multigrid method. Particularly, we use a geometric version of the Multigrid technique. The proposal is based on a modification of the operators in some levels of the Multigrid algorithm in order to achieve both, the advantages of Multigrid for linear systems resolution and the solution improvement that could be achieved with the Composite Mesh strategy. Several elliptic test problems with analytical solution are presented, where the convergence rates are analyzed and the decrease in discretization errors is quantified.