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

  • Svetlana TikhovskayaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)

Abstract

A two-dimensional linear elliptic equation with 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 parameter \(\varepsilon \). 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 the \(\varepsilon \)-uniform accuracy of the difference scheme. In this paper multigrid algorithm of the same structure is studied. We construct 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 the \(\varepsilon \)-uniform accuracy of the difference scheme by two orders. The different components of a cascadic multigrid method are studied. The results of some numerical experiments are discussed.

Keywords

Singularly perturbed elliptic problem Regular boundary layers Difference scheme Shishkin mesh \(\varepsilon \)-uniform accuracy Cascadic multigrid method Richardson extrapolation 

Notes

Acknowledgments

Research has been supported by the program of fundamental scientific researches of the SB RAS No I.1.3., project No 0314-2016-0009.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics SB RASOmskRussia

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