# zbMATH — the first resource for mathematics

A technique to prove parameter-uniform convergence for a singularly perturbed convection-diffusion equation. (English) Zbl 1117.65145
Summary: A priori parameter explicit bounds on the solution of singularly perturbed elliptic problems of convection-diffusion type are established. Regular exponential boundary layers can appear in the solution. These bounds on the solutions and its derivatives are obtained using a suitable decomposition of the solution into regular and layer components. By introducing extensions of the coefficients to a larger domain, artificial compatibility conditions are not imposed in the derivation of these decompositions.

##### MSC:
 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 35B25 Singular perturbations in context of PDEs 65N15 Error bounds for boundary value problems involving PDEs 65N06 Finite difference methods for boundary value problems involving PDEs
Full Text:
##### References:
 [1] Bobisud, L., Second-order linear parabolic equations with a small parameter, Arch. rational mech. anal., 27, 385-397, (1967) · Zbl 0153.14203 [2] Dobrowolski, M.; Roos, H.-G., A priori estimates for the solution of convection-diffusion problems and interpolation on shishkin meshes, Z. anal. anwendungen, 16, 1001-1012, (1997) · Zbl 0892.35014 [3] Farrell, P.A.; Hegarty, A.F.; Miller, J.J.H.; O’Riordan, E.; Shishkin, G.I., Robust computational techniques for boundary layers, (2000), Chapman & Hall/CRC Press Boca Raton, USA · Zbl 0964.65083 [4] Han, H.; Kellogg, R.B., Differentiability properties of solutions of the equation $$- \varepsilon^2 ▵ u + \mathit{ru} = f(x, y)$$ in a square, SIAM J. math. anal., 21, 394-408, (1990) · Zbl 0732.35020 [5] Hegarty, A.F.; Miller, J.J.H.; O’Riordan, E.; Shishkin, G.I., On a novel mesh for the regular boundary layers arising in advection-dominated transport in two dimensions, Comm. numer. methods eng., 11, 435-441, (1995) · Zbl 0824.65100 [6] Il’in, A.M., Matching of asymptotic expansions of solutions of boundary value problems, (1992), American Mathematical Society Providence, RI · Zbl 0754.34002 [7] Kellogg, R.B.; Stynes, M., Corner singularities and boundary layers in a simple convection diffusion problem, J. differential equations, 213, 81-120, (2005) · Zbl 1159.35309 [8] Ladyzhenskaya, O.A.; Ural’tseva, N.N., Linear and quasilinear elliptic equations, (1968), Academic Press New York, London · Zbl 0164.13002 [9] 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 [10] Miller, J.J.H.; O’Riordan, E.; Shishkin, G.I., Fitted numerical methods for singular perturbation problems, (1996), World Scientific Singapore · Zbl 0945.65521 [11] Roos, H.-G., 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 [12] Shih, S.; Kellogg, R.B., Asymptotic analysis of a singular perturbation problem, SIAM J. math. anal., 18, 1467-1511, (1987) · Zbl 0642.35006 [13] G.I. Shishkin, Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equations, Russian Academy of Sciences, Ural section, Ekaterinburg, 1992 (in Russian). [14] Volkov, E.A., Differentiability properties of solutions of boundary value problems for the Laplace and Poisson equations, Proc. Steklov inst. math., 77, 101-126, (1965) · Zbl 0162.16602
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.