Abstract
A singularly perturbed elliptic convection–diffusion equation with a perturbation parameter ε (ε ∈ (0, 1]) is considered on a rectangle. As applied to this equation, a standard finite difference scheme on a uniform grid is studied under computer perturbations. This scheme is not ε-uniformly stable with respect to perturbations. The conditions imposed on a “computing system” are established under which a converging standard scheme (referred to as a computer difference scheme) remains stable.
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References
G. I. Shishkin and L. P. Shishkina, Difference Methods for Singular Perturbation Problems (CRC, Boca Raton, FL, 2009).
A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).
G. I. Shishkin, Dokl. Math. 87 (1), 107–109 (2013).
G. I. Shishkin, Dokl. Math. 91 (3), 273–276 (2015).
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Original Russian Text © G.I. Shishkin, 2016, published in Doklady Akademii Nauk, 2016, Vol. 467, No. 3, pp. 271–274.
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Shishkin, G.I. Standard finite difference scheme for a singularly perturbed elliptic convection–diffusion equation on a rectangle under computer perturbations. Dokl. Math. 93, 179–182 (2016). https://doi.org/10.1134/S1064562416020150
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DOI: https://doi.org/10.1134/S1064562416020150