Solving a Singularly Perturbed Elliptic Problem by a Multigrid Algorithm with Richardson Extrapolation

Abstract

A two-dimensional linear elliptic equation with regular boundary layers is considered. It is solved by using an upwind difference scheme on the Shishkin mesh with the property of uniform convergence with respect to small parameter \(\varepsilon \). It is known that the application of multigrid methods leads to essential reduction of the number of arithmetical operations. Earlier the two-grid method with the application Richardson extrapolation to increase the \(\varepsilon \)-uniform accuracy of the difference scheme is investigated. In this paper the multigrid algorithm of the same structure is considered. The application of the Richardson extrapolation with the usage of numerical solutions on all of meshes leads to increase the \(\varepsilon \)-uniform accuracy of the difference scheme by two orders. Also we construct a better initial guess on refined mesh, using numerical solutions on coarse meshes. The results of some numerical experiments are discussed.