The necessity of Shishkin decompositions

Applied Mathematics Letters

Volumn 14, Issue 7, 2001, Pages 891-896

  Linss T ^a  

^a DRESDEN UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Convection diffusion problems; Interpolation error; Layer adapted meshes; Shishkin decomposition; Singular perturbation

Indexed keywords

EID: 4243906064     PISSN: 08939659     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0893-9659(01)00061-1     Document Type: Article

References (15)

1. Kellogg R.B., Tsan A. Analysis of some difference approximations for a singular perturbation problem without turning points. Math. Comp. 32:1978;1025-1039.
2. Roos H.-G., Stynes M., Tobiska L. Numerical Methods for Singularly Perturbed Differential Equations. 24:1996;Springer Series in Computational Mathematics, Springer-Verlag, Berlin.
3. Miller J.J.H., O'Riordan E., Shishkin G.I. Solution of Singularly Perturbed Problems with ε-Uniform Numerical Methods - Introduction to the Theory of Linear Problems in One and Two Dimensions. 1996;World Scientific, Singapore.
4. Shishkin G.I. Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equations. 1992;Russian Academy of Sciences, Ural Section, Ekaterinburg. (in Russian).
5. G.M. Beckett and J.A. Mackenzie, Convergence analysis of finite difference approximations on equidistributed grids to a singularly perturbed boundary value problem, Appl. Numer. Math. (to appear).
6. Roos H.-G., Linß T. Sufficient conditions for uniform convergence on layer adapted grids. Computing. 63:1999;27-45.
7. Stynes M., Roos H.-G. The midpoint upwind scheme. Appl. Numer. Math. 32:1997;291-308.
8. Stynes M., Tobiska L. A finite difference analysis of a streamline diffusion method on a Shishkin mesh. Numer. Algorithms. 18:1998;337-360.
9. Linß T. Uniform pointwise convergence of finite difference schemes using grid equidistribution. November 1999;Institut für Numerische Mathematic, Technische Universität Dresden. Preprint MATH-NM-11-99.
10. Stynes M., O'Riordan E. A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem. J. Math. Anal. Applic. 214:1997;36-54.
11. T. Linß, Uniform superconvergence of a Galerkin finite element method on Shishkin-type meshes for a linear convection-diffusion problem, Numer. Methods Partial Differential Equations (to appear).
12. de Boor C. Good approximation by splines with variable knots. Meir A., Sharma A. Spline Functions and Approximation Theory, Proceedings of the Symposium. May 29-June 1, 1972;University of Alberta, Edmonton. Birkhäuser, Basel, (1973).
13. Linß T., Stynes M. Asymptotic analysis and Shishkin-type decompositions for an elliptic linear convection-diffusion problem. 1998;Department of Mathematics, National University of Ireland, Cork. Preprint 1998-4.
14. Bakhvalov N.S. Towards optimization of methods for solving boundary value problems in the presence of a boundary layer. (in Russian) Zh. Vychisl. Mat. Mat. Fiz. 9:1969;841-859.
15. Dobrowolski M., Roos H.-G. A priori estimates for the solution of convection-diffusion problems and interpolation on Shishkin meshes. Z. Anal. Anwendungen. 16:1997;1001-1012.