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 Sib. Zh. Vychisl. Mat., 2002, Volume 5, Number 1, Pages 71–92 (Mi sjvm240)

Grid approximations with an improved rate of convergence for singularly perturbed elliptic equations in domains with characteristic boundaries

G. I. Shishkin

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: On a rectangle, we consider the Dirichlet problem for singularly perturbed elliptic equations with convective terms in the case of characteristics of the reduced equations which are parallel to the sides. For such convection-diffusion problems the uniform (with respect to the perturbation parameter $\varepsilon$) convergence rate of the well-known special schemes on piecewise uniform meshes is of order not higher than one (in the uniform $L_{\infty}$-norm). For the above problem, based on asymptotic expansions of the solutions, we construct schemes that converge $\varepsilon$-uniformly with the rate $\mathscr O(N^{-2}\ln^2N)$, where $N$ defines the number of mesh points with respect to each variable. For not too small values of the parameter we apply classical finite difference approximations on piecewise uniform meshes condensing in boundary layers; for small values of the parameter we use approximations of auxiliary problems, which describe the main terms of asymptotic representation of the solution in a neighborhood of the boundary layer and outside of it. Note that the computation of solutions of the constructed difference scheme is simplified for sufficiently small values of the parameter $\varepsilon$.

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Bibliographic databases:
UDC: 519.632.4
Revised: 28.03.2001

Citation: G. I. Shishkin, “Grid approximations with an improved rate of convergence for singularly perturbed elliptic equations in domains with characteristic boundaries”, Sib. Zh. Vychisl. Mat., 5:1 (2002), 71–92

Citation in format AMSBIB
\Bibitem{Shi02} \by G.~I.~Shishkin \paper Grid approximations with an improved rate of convergence for singularly perturbed elliptic equations in domains with characteristic boundaries \jour Sib. Zh. Vychisl. Mat. \yr 2002 \vol 5 \issue 1 \pages 71--92 \mathnet{http://mi.mathnet.ru/sjvm240} \zmath{https://zbmath.org/?q=an:1027.65138} 

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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. Shishkin G.I., Shishkina L.P., “A higher-order Richardson method for a quasilinear singularly perturbed elliptic reaction-diffusion equation”, Differ. Equ., 41:7 (2005), 1030–1039
2. G. I. Shishkin, “Grid approximation of a singularly perturbed elliptic equation with convective terms in the presence of various boundary layers”, Comput. Math. Math. Phys., 45:1 (2005), 104–119
3. G. I. Shishkin, “Grid approximation of the domain and solution decomposition method with improved convergence rate for singularly perturbed elliptic equations in domains with characteristic boundaries”, Comput. Math. Math. Phys., 45:7 (2005), 1155–1171
4. G. I. Shishkin, “Metod povyshennoi tochnosti dlya kvazilineinogo singulyarno vozmuschennogo ellipticheskogo uravneniya konvektsii-diffuzii”, Sib. zhurn. vychisl. matem., 9:1 (2006), 81–108
5. Shishkin G.I., “Grid approximation of a singularly perturbed elliptic convection-diffusion equation in an unbounded domain”, Russian J. Numer. Anal. Math. Modelling, 21:1 (2006), 67–94
6. G. I. Shishkin, “A method of asymptotic constructions of improved accuracy for a quasilinear singularly perturbed parabolic convection-diffusion equation”, Comput. Math. Math. Phys., 46:2 (2006), 231–250
7. Shishkin G.I., Shishkina L.P., “The Richardson extrapolation technique for quasilinear parabolic singularly perturbed convection-diffusion equations”, International Workshop on Multi-Rate Processes and Hysteresis, Journal of Physics Conference Series, 55, 2006, 203–213
8. G. I. Shishkin, L. P. Shishkina, “Improved difference scheme of the solution decomposition method for a singularly perturbed reaction-diffusion equation”, Proc. Steklov Inst. Math. (Suppl.), 272, suppl. 1 (2011), S197–S214
9. Shishkin G.I., “Difference scheme of the solution decomposition method for a singularly perturbed parabolic reaction-diffusion equation”, Russian J Numer Anal Math Modelling, 25:3 (2010), 261–278
10. G. I. Shishkin, L. P. Shishkina, “A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation”, Comput. Math. Math. Phys., 50:12 (2010), 2003–2022
11. 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
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