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Approximation of systems of singularly perturbed elliptic reaction-diffusion equations with two parameters

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Abstract

In a rectangle, the Dirichlet problem for a system of two singularly perturbed elliptic reaction-diffusion equations is considered. The higher order derivatives of the ith equation are multiplied by the perturbation parameter ɛ 2 i (i = 1, 2). The parameters ɛi take arbitrary values in the half-open interval (0, 1]. When the vector parameter ɛ = (ɛ1, ɛ2) vanishes, the system of elliptic equations degenerates into a system of algebraic equations. When the components ɛ1 and (or) ɛ2 tend to zero, a double boundary layer with the characteristic width ɛ1 and ɛ2 appears in the vicinity of the boundary. Using the grid refinement technique and the classical finite difference approximations of the boundary value problem, special difference schemes that converge ɛ-uniformly at the rate of O(N −2ln2 N) are constructed, where N = min N s, N s + 1 is the number of mesh points on the axis x s.

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Authors and Affiliations

  1. Institute of Mathematics and Mechanics, Ural Division, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620219, Russia

    G. I. Shishkin

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Original Russian Text © G.I. Shishkin, 2007, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2007, Vol. 47, No. 5, pp. 835–866.

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Shishkin, G.I. Approximation of systems of singularly perturbed elliptic reaction-diffusion equations with two parameters. Comput. Math. and Math. Phys. 47, 797–828 (2007). https://doi.org/10.1134/S0965542507050077

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  • DOI: https://doi.org/10.1134/S0965542507050077

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