Preconditioner for a discrete Laplace operator on a condensed grid

Abstract

In this paper, it is proved that the discrete Laplace operator approximating a Dirichlet boundary value problem for a Poisson equation by a finite element method with piecewise-linear functions on an evenly condensed grid that is topologically equivalent to a rectangular grid (i.e., obtained by shifting the rectangular grid nodes) is equivalent, in the range, to the operator of a 5-point difference scheme on a uniform grid.