A Cascadic Multigrid Algorithm on the Shishkin Mesh for a Singularly Perturbed Elliptic Problem

Abstract

A two-dimensional linear elliptic problem with parabolic and regular boundary layers is considered in the unit square. It is solved by using an upwind difference scheme on the Shishkin mesh which converges uniformly with respect to a small perturbation parameter. The scheme is resolved based on an iterative method. It is known that the application of multigrid methods leads to essential reduction of arithmetical operations amount. Earlier we investigated the cascadic two-grid method with the application of Richardson extrapolation to increase accuracy of the difference scheme uniformly with respect to a perturbation parameter for a linear elliptic equation with parabolic and regular boundary layers and the cascadic multigrid method with the application of Richardson extrapolation for a linear elliptic equation with regular boundary layers. In this paper a cascadic multigrid algorithm of the same structure is studied. We also used an interpolation formula uniform with respect to a perturbation parameter and an extrapolation of initial guess using numerical solutions on two coarse meshes to reduce the arithmetical operations amount. The application of the Richardson extrapolation method based on numerical solutions on the last three meshes leads to increase accuracy of the difference scheme by two orders uniformly with respect to a perturbation parameter. We compare the proposed cascadic multigrid method with a multigrid method with V-cycle with a special restriction operator. The results of some numerical experiments are discussed.