An algebraic study of a local multigrid method for variational problems

摘要

A two-level iterative method for solving systems of equations arising from local refinement of finite element meshes is proposed. The smoothing steps consists of Gauss-Seidel relaxations applied only to the local mesh. A direct solver is required on the global coarse mesh. We analyze this scheme by modifying the standard algebraic convergence theory. The estimates of the convergence factor do not require any regularity assumption. The estimates can be evaluated element-by-element. The evaluated estimates are compared with the actual convergence factor and with another bound in several numerical examples. The theory also applies to a method with a direct solver on the local fine mesh. In this case we have obtained very sharp (and in fact optimal) estimates of the convergence factors.