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 Zh. Vychisl. Mat. Mat. Fiz., 2008, Volume 48, Number 1, Pages 90–114 (Mi zvmmf197)

Uniform grid approximation of nonsmooth solutions to the mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a rectangle

V. B. Andreev

Faculty of Computational Mathematics and Cybernetics, Moscow State University, Leninskie gory, Moscow, 119992, Russia

Abstract: A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a square is considered. A Neumann condition is specified on one side of the square, and a Dirichlet condition is set on the other three. It is assumed that the coefficient of the equation, its right-hand side, and the boundary values of the desired solution or its normal derivative on the sides of the square are smooth enough to ensure the required smoothness of the solution in a closed domain outside the neighborhoods of the corner points. No compatibility conditions are assumed to hold at the corner points. Under these assumptions, the desired solution in the entire closed domain is of limited smoothness: it belongs only to the Hölder class $C^\mu$, where $\mu\in(0,1)$ is arbitrary. In the domain, a nonuniform rectangular mesh is introduced that is refined in the boundary domain and depends on a small parameter. The numerical solution to the problem is based on the classical five-point approximation of the equation and a four-point approximation of the Neumann boundary condition. A mesh refinement rule is described under which the approximate solution converges to the exact one uniformly with respect to the small parameter in the $L_\infty^h$ norm. The convergence rate is $O(N^{-2}\ln^2N)$, where $N$ is the number of mesh nodes in each coordinate direction. The parameter-uniform convergence of difference schemes for mixed problems without compatibility conditions at corner points was not previously analyzed.

Key words: singularly perturbed reaction-diffusion equation, mixed boundary value problem, finite-difference method, refined meshes, uniform convergence.

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English version:
Computational Mathematics and Mathematical Physics, 2008, 48:1, 85–108

Bibliographic databases:

UDC: 519.632.4

Citation: 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”, Zh. Vychisl. Mat. Mat. Fiz., 48:1 (2008), 90–114; Comput. Math. Math. Phys., 48:1 (2008), 85–108

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Andreev V.B., “Uniform mesh approximation to nonsmooth solutions of a singularly perturbed convection-diffusion equation in a rectangle”, Differ. Equ., 45:7 (2009), 973–982
2. Ershova T.Ya., “Solution of the Dirichlet problem for a singularly perturbed reaction-diffusion equation in a square on a Bakhvalov grid”, Mosc. Univ. Comput. Math. Cybern., 33:4 (2009), 171–180
3. Kopteva N., O'Riordan E., “Shishkin meshes in the numerical solution of singularly perturbed differential equations”, Int. J. Numer. Anal. Model., 7:3 (2010), 393–415
4. Andreev V.B., “Pointwise approximation of corner singularities for singularly perturbed elliptic problems with characteristic layers”, Int. J. Numer. Anal. Model., 7:3 (2010), 416–427
5. Kadalbajoo M.K., Gupta V., “A brief survey on numerical methods for solving singularly perturbed problems”, Appl. Math. Comput., 217:8 (2010), 3641–3716
6. U. H. Zhemuhov, “Uniform grid approximation of nonsmooth solutions with improved convergence for a singularly perturbed convection-diffusion equation with characteristic layers”, Comput. Math. Math. Phys., 52:9 (2012), 1239–1259
7. Ershova T.Ya., “Smeshannaya kraevaya zadacha dlya singulyarno vozmuschennogo uravneniya reaktsii-diffuzii v $l$-obraznoi oblasti”, Vestn. Mosk. un-ta. Ser. 15: Vychislitelnaya matematika i kibernetika, 3 (2012), 3–12
8. V. B. Andreev, “Estimating the smoothness of the regular component of the solution to a one-dimensional singularly perturbed convection-diffusion equation”, Comput. Math. Math. Phys., 55:1 (2015), 19–30
9. V. B. Andreev, “Hölder estimates for the regular component of the solution to a singularly perturbed convection-diffusion equation”, Comput. Math. Math. Phys., 57:12 (2017), 1935–1972
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