×

Corner singularities and boundary layers in a simple convection-diffusion problem. (English) Zbl 1159.35309

Summary: A singularly perturbed convection-diffusion problem posed on the unit square is considered. Its solution may have exponential and parabolic boundary layers, and corner singularities may also be present. Pointwise bounds on the solution and its derivatives are derived. The dependence of these bounds on the small diffusion coefficient, on the regularity of the data, and on the compatibility of the data at the corners of the domain are all made explicit. The bounds are derived by decomposing the solution into a sum of solutions of elliptic boundary-value problems posed on half-planes, then analyzing these simpler problems.

MSC:

35B25 Singular perturbations in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1965.; M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1965.
[2] Grisvard, P., Elliptic Problems in Nonsmooth Domains (1985), Pitman: Pitman London · Zbl 0695.35060
[3] Han, H.; Kellogg, R. B., Differentiability properties of solutions of the equation \(- \epsilon^2 \Delta u + ru = f\) in a square, SIAM J. Math. Anal., 21, 394-408 (1990) · Zbl 0732.35020
[4] Il’in, A. M., Matching of Asymptotic Expansions of Solutions of Boundary Value Problems (1992), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0754.34002
[5] Kellogg, R. B., Corner singularities and singular perturbations, Ann. dell’Univ. Ferrara, 47, 7, 177-206 (2001) · Zbl 1119.35318
[6] Kellogg, R. B.; Tsan, A., Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comp., 34, 1025-1039 (1978) · Zbl 0418.65040
[7] Linss, T.; Stynes, M., Asymptotic analysis and Shishkin-type decomposition for an elliptic convection-diffusion problem, J. Math. Anal. Appl., 261, 604-632 (2001) · Zbl 1200.35046
[8] Miller, J. J.H.; O’Riordan, E.; Shishkin, G. I., Fitted mesh methods for the singularly perturbed reaction-diffusion problem, (Minchev, E., Proceedings of the Vth International Conference on Numerical Analysis, 1996 (1997), Plovdiv, Bulgaria, Academic Publications: Plovdiv, Bulgaria, Academic Publications New York), 99-105 · Zbl 0814.65082
[9] Hans-Görg Roos, Optimal convergence of basic schemes for elliptic boundary value problems with strong parabolic layers, J. Math. Anal. Appl., 267, 194-208 (2002) · Zbl 1051.65109
[10] Shagi-di Shih; Kellogg, R. B., Asymptotic analysis of a singular perturbation problem, SIAM J. Math. Anal., 18, 1467-1511 (1987) · Zbl 0642.35006
[11] G.I. Shishkin, Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equations, Russian Academy of Sciences, Ural Section, Ekaterinburg, 1992 (in Russian).; G.I. Shishkin, Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equations, Russian Academy of Sciences, Ural Section, Ekaterinburg, 1992 (in Russian). · Zbl 1397.65005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.