Multigrid methods and adaptive refinement techniques in elliptic problems by finite element methods

Published in Comput. Methods Appl. Mech. Engrg. Vol. 93(1), pp. 13-30, 1991
doi: 10.1016/0045-7825(91)90113-K

Abstract

Multigrid and adaptive refinement techniques are powerful tools in the resolution of problems arising from computational mechanics. In structured grids the number of numerical operations needed in the resolution via the multigrid method is of order N, the number of degrees of freedom, regardless the dimensionality of the problem. In this work a multigrid technique, which works on unstructured meshes, is described. These meshes are obtained from a user defined tesselation through bisection refinement upon any element. Original techniques for nodes numbering, irregular nodes detection and nodes classification are developed in view of multigrid method implementation. The error indicator used in the refinement process is based on truncation error estimates. The refinement strategy is oriented toward error reduction at constant computational effort. Finally several numerical examples are presented.