Maximum norm a posteriori error estimates for a one-dimensional convection-diffusion problem

SIAM Journal on Numerical Analysis

Volumn 39, Issue 2, 2002, Pages 423-441

  Kopteva, Natalia ^a  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

A posteriori error estimate; Convection diffusion; Finite difference scheme; Grid equidistribution; Layer adapted mesh; Quasi linear problem; Singular perturbation; Upwind scheme

Indexed keywords

EID: 0035322506     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142900368642     Document Type: Article

References (31)

1. R. A. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.
2. V. B. ANDREEV AND N. V. KOPTEVA, A study of difference schemes with the first derivative approximated by a central difference ratio, Comput. Math. Math. Phys., 36 (1996), pp. 1065-1078.
3. V. B. ANDREEV AND N. B. KOPTEVA, On the convergence, uniform with respect to a small parameter, of monotone three-point difference approximations, Differ. Uravn., 34 (1998), pp. 921-929.
4. I. BABUŠKA AND W. C. RHEINBOLDT, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal., 15 (1978), pp. 736-754.
5. N. S. BAKHVALOV, Towards optimization of methods for solving boundary value problems in the presence of a boundary layer, Zh. Vychisl. Mat. Mat. Fiz., 9 (1969), pp. 841-859 (in Russian).
6. 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., 35 (2000), pp. 87-109.
7. A. E. BERGER, A conservative uniformly accurate difference method for a singular perturbation problem in conservative form, SIAM J. Numer. Anal., 23 (1986), pp. 1241-1253.
8. A. L. GONTCHAROV AND I. W. FRJAZINOV, A ten-point difference scheme for the solution of the Navier-Stokes equations, Zh. Vychisl. Mat. Mat. Fiz., 28 (1988), pp. 867-878 (in Russian).
9. D. H. GRIFFEL, Applied Functional Analysis, Ellis Horwood Ltd., Chichester, 1981.
10. B. F. GROMOV AND V. S. PETRISHCHEV, On numerical solution to two-dimensional incompressible viscous fluid problems, in Proceedings of All-Union Seminar on Computational Viscous Fluid Mechanics, Nauka, Novosibirsk, 1969 (in Russian).
11. W. DÖRFLER, Uniformly Convergent Finite-Element Methods for Singularly Perturbed Convection-Diffusion Equations, Habilitationsschrift, Albert-Ludwig-Universität, Freiburg, Germany, 1998.
12. N. V. KOPTEVA, Uniform convergence with respect to a small parameter of a four-point scheme for the one-dimensional stationary convection-diffusion equation, Differential Equations, 32 (1996), pp. 958-964.
13. N. V. KOPTEVA, On the uniform with respect to a small parameter convergence of the central difference scheme on condensing meshes, Comput. Math. Math. Phys., 39 (1999), pp. 1669-1685.
14. N. KOPTEVA, Uniform pointwise convergence of difference schemes for convection-diffusion problems on layer-adapted meshes, Computing, 66 (2001), pp. 179-197.
15. N. KOPTEVA AND T. LINß, Uniform second-order pointwise convergence of a central difference approximation for a quasilinear convection-diffusion problem, J. Comput. Appl. Math., to appear.
16. N. KOPTEVA AND M. STYNES, A Robust Adaptive Method for a Quasilinear One-Dimensional Convection-Diffusion Problem, Preprint 2001-2, Department of Mathematics, National University of Ireland, Cork, Ireland.
17. T. LINß, Uniform pointwise convergence of finite difference schemes using grid equidistribution, Computing, 66 (2001), pp. 27-39.
18. J. LORENZ, Stability and monotonicity properties of stiff quasilinear boundary problems, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat., 12 (1982), pp. 151-175.
19. J. A. MACKENZIE, The efficient generation of simple two-dimensional adaptive grids, SIAM J. Sci. Comput., 19 (1998), pp. 1340-1365.
20. J. A. MACKENZIE, Uniform convergence analysis of an upwind finite-difference approximation of a convection-diffusion boundary value problem on an adaptive grid, IMA J. Numer. Anal., 19 (1999), pp. 233-249.
21. R. E. O'MALLEY, JR., Singular Perturbation Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991.
22. J. J. H. MILLER, E. O'RIORDAN, AND G. I. SHISHKIN, Solution of Singularly Perturbed Problems with ε-uniform Numerical Methods - Introduction to the Theory of Linear Problems in One and Two Dimensions, World Scientific, Singapore, 1996.
23. Y. QIU AND D. M. SLOAN, Analysis of difference approximations to a singularly perturbed two-point boundary value problem on an adaptively generated grid, J. Comput. Appl. Math., 101 (1999), pp. 1-25.
24. Y. QIU, D. M. SLOAN, AND T. TANG, Numerical solution of a singularly perturbed two-point boundary value problem using equidistribution: Analysis of convergence, J. Comput. Appl. Math., 116 (2000), pp. 121-143.
25. H.-G. Roos, Layer-adapted grids for singular perturbation problems, ZAMM Z. Angew. Math. Mech., 78 (1998), pp. 291-309.
26. H.-G. ROOS, M. STYNES, AND L. TOBISKA, Numerical Methods for Singularly Perturbed Differential Equations, Springer-Verlag, Berlin, 1996.
27. A. A. SAMARSKI, Theory of Difference Schemes, Nauka, Moscow, 1989 (in Russian).
28. G. I. SHISHKIN, Grid Approximation of Singularly Perturbed Elliptic and Parabolic Equations, Ural Branch of Russian Academy of Sciences, Ekaterinburg, Russia, 1992 (in Russian).
29. W. G. SZYMCZAK AND I. BABUŠKA, Adaptivity and error estimation for the finite element method applied to convection diffusion problems, SIAM J. Numer. Anal., 21 (1984), pp. 910-954.
30. R. VERFÜRTH, A posteriori error estimation and adaptive mesh-refinement techniques, J. Comput. Appl. Math., 50 (1994), pp. 67-83.
31. R. VULANOVIĆ, A uniform numerical method for quasilinear singular perturbation problems without turning points, Computing, 41 (1989), pp. 97-106.