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

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 perturbation parameter. The scheme is resolved based on an iterative meth...

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Vydáno v:Journal of physics. Conference series Ročník 2182; číslo 1; s. 12034 - 12042
Hlavní autor: Tikhovskaya, S V
Médium: Journal Article
Jazyk:angličtina
Vydáno: Bristol IOP Publishing 01.03.2022
Témata:
Algorithms
Boundary layers
Extrapolation
Grid method
Interpolation
Iterative methods
Multigrid methods
Parameters
Perturbation
Physics
ISSN:1742-6588, 1742-6596
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Shrnutí: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 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 by an order uniform with respect to a perturbation parameter, using an interpolation formula uniform with respect to a perturbation parameter. In this paper a cascadic 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 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. The results of some numerical experiments are discussed.
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ISSN:1742-6588
1742-6596
DOI:10.1088/1742-6596/2182/1/012034