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A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem. (English) Zbl 0917.65088
Two results on convergence uniform in the singular perturbation parameter are given for a singularly perturbed linear elliptic boundary value problem in two dimensions. Their error bounds are of order (i) \((\ln N/N)\) in a global energy norm and (ii) \((\ln N)^{3/2}/\sqrt N\) pointwise near the outflow boundary.

MSC:
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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[1] Andreev, W. B.; Kopteva, N. W., On the investigation of difference schemes with a central approximation of the first order derivative, Zh. Vychisl. Mat. i Mat. Fiz., 36, 101-117, (1996)
[2] Bakhvalov, N. S., Towards optimization of methods for solving boundary value problems in the presence of boundary layers, Zh. Vychisl. Mat. i Mat. Fiz., 9, 841-859, (1969)
[3] Han, H.; Kellogg, R. B., Differentiability properties of solutions of the equation −ε^{2}uruf, SIAM J. Math. Anal., 21, 394-408, (1990) · Zbl 0732.35020
[4] Miller, J. J.H.; O’Riordan, E.; Shishkin, G. I., Fitted Numerical Methods for Singular Perturbation Problems—Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, (1996), World Scientific Singapore · Zbl 0915.65097
[5] E. O’Riordan, M. Stynes, 1989, A uniformly convergent difference scheme for an elliptic singular perturbation problem, Proc. Conf. on Discretization Methods in Singular Perturbations and Flow Problems, L. Tobiska, 48, 55, Technical University “Otto von Guericke” Magdeburg, G.D.R.
[6] O’Riordan, E.; Stynes, M., A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions, Math. Comp., 57, 47-62, (1991) · Zbl 0733.65063
[7] Roos, H.-G., Necessary convergence conditions for upwind schemes in the two-dimensional case, Internat. J. Numer. Methods Engrg., 21, 1459-1469, (1985) · Zbl 0578.65098
[8] H.-G. Roos, 1996, A Priori Estimates, Asymptotic Expansions and Shishkin Decompositions, Technical University of Dresden
[9] Roos, H.-G.; Adam, D.; Felgenhauer, A., A novel uniformly convergent finite element method in two dimensions, J. Math. Anal. Appl., 201, 711-755, (1996) · Zbl 0859.65118
[10] Roos, H.-G.; Stynes, M.; Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations—Convection-Diffusion and Flow Problems, (1996), Springer-Verlag Heidelberg · Zbl 0844.65075
[11] Shishkin, G. I., Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equations, (1992), Russian Academy of SciencesUral Section Ekaterinburg · Zbl 0808.65102
[12] Sun, G.; Stynes, M., Finite-element methods for singularly perturbed high-order elliptic two-point boundary value problems. II. convection-diffusion-type problems, IMA J. Numer. Anal., 15, 197-219, (1995) · Zbl 0824.65077
[13] Vulanović, R., Non-equidistant finite difference methods for elliptic singular perturbation problems, (Miller, J. J.H., Computational Methods for Boundary and Interior Layers in Several Dimensions, (1991), Boole Dublin), 203-223 · Zbl 0769.65074
[14] Wait, R.; Mitchell, A. R., Finite Element Analysis and Applications, (1985), Wiley New York · Zbl 0577.65093
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