Advanced Mathematics, Computations and Applications 2015

October 19-23, 2015, Akademgorodok, Novosibirsk, Russia

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Tikhovskaya S.  

Multigrid algorithm for solution of singularly perturbed elliptic problem

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 [1] with the property of uniform convergence with respect to a small parameter. It is known that the application of multigrid methods leads to essential reduction of the number of arithmetical operations. In [2], the two-grid method with the application Richardson extrapolation to increase the accuracy of the difference scheme is investigated. It is shown that in case of the auxiliary mesh with half the number of nodes than the original leads to increase the accuracy of the difference scheme by an order. In this paper the multigrid algorithm of the same structure is investigated. For ease of comparison we used only one additional auxiliary grid with the number of nodes is four times less than the original. The application of the Richardson extrapolation method with the usage of numerical solutions of all meshes leads to increase the accuracy of the difference scheme by two orders.
The project has been partially supported by grants 15-01-06584

REFERENCES
1. Shishkin G.I. Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations. Yekaterinburg: UB RAS, 1992 [in Russian].
2. Tikhovskaya S.V. A two-grid method for an elliptic equation with boundary layers on a Shishkin mesh // Lobachevskii Journal of Mathematics. 2014. Vol. 35, № 4. P. 391–397.