Abstract
A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in an L-shaped domain is considered for when the solution has singularities at the corners of the domain. The densification of the Shishkin mesh near the inner corner where different boundary conditions meet is such that the solution obtained by the classical five-point difference scheme converges to the solution of the initial problem in the mesh norm L h∞ uniformly with respect to the small parameter with almost second order, i.e., as a smooth solution. Numerical analysis confirms the theoretical result.
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References
E. A. Volkov, “On the Regular Compound Grid Method for the Laplace Equation in Polygons,” Tr. Mat. Inst. Acad. Nauk 140, 68–102 (1976).
H. Han and R. Kellogg, “Differentiability Properties of Solutions of the Equation −ɛ2Δu + ru = f(x, y) in a Square,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 21, 394–408 (1990).
G. I. Shishkin, Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations (Ural. Otd. Ross. Akad. Nauk, Yekaterinburg, 1992) [in Russian].
E. A. Volkov, “On Differential Properties of Solutions to the Boundary-Value Problems for the Laplace and Poisson Equations in a Rectangle,” Tr. Mat. Inst. Acad. Nauk 77, 89–112 (1965).
V. B. Andreev, “Uniform Grid Approximation of Nonsmooth Solutions to the Mixed Boundary Value Problem for a Singularly Perturbed Reaction-Diffusion Equation in a Rectangle,” Comput. Math. Math. Phys. 48, 85–108 (2008).
V. Andreev and N. V. Kopteva, “Pointwise Approximation of Corner Singularities for a Singularly Perturbed Reaction-Diffusion Equation in An L-Shaped Domain,” Math. Comput. 77(264), 2125–2139 (2008).
O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasi-Linear Equations of Elliptic Type (Nauka, Moscow, 1973) [in Russian].
N. Wigley, “Asymptotic Expansions at a Corner of Solutions of Mixed Boundary Value Problems,” J. Math. and Mech. 13, 549–576 (1964).
A. A. Samarskii and V. B. Andreev, Difference Methods for Elliptic Equations (Nauka, Moscow, 1976) [in Russian].
V. B. Andreev, “Pointwise Approximation of Corner Singularities for Singularly Perturbed Elliptic Problems with Characteristic Layers,” Int. J. Numer. Analysis Modeling 7, 416–427 (2010).
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Original Russian Text © T.Ya. Ershova, 2012, published in Vestnik Moskovskogo Universiteta. Vychislitel’naya Matematika i Kibernetika, 2012, No. 3, pp. 3–12.
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Ershova, T.Y. A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in an L-shaped domain. MoscowUniv.Comput.Math.Cybern. 36, 109–119 (2012). https://doi.org/10.3103/S0278641912030028
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DOI: https://doi.org/10.3103/S0278641912030028