A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations

Numerische Mathematik

Volumn 87, Issue 4, 2001, Pages 737-761

  Ohlberger, M ^a  

^a UNIVERSITY OF FREIBURG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035585644     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/PL00005431     Document Type: Article

References (24)

1. P. Angot, V. Dolejší, M. Feistauer, and J. Felcman. Analysis of a combined barycentric finite volume - finite element method for nonlinear convection diffusion problems. Appl. Math. Praha 43(4), 263-311 (1998)
2. C. Chainais-Hillairet. Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimates. Mathematical Modelling and Numerical Analysis 33, 129-156 (1999)
3. B. Cockburn, F. Coquel, and P.G. Lefloch. An error estimate for finite volume methods for multidimensional conservation laws. Math. Comput. 63, 77-103 (1994)
4. B. Cockburn and H. Gau. A posteriori error estimates for general numerical methods for scalar conservation laws. Comput. Appl. Math. 14, 37-47 (1995)
5. B. Cockburn and P.A. Gremaud. Error estimates for finite element methods for scalar conservation laws. SIAM J. Num. Anal. 33, 522-554 (1996)
6. W. Dörfler. Uniformly Convergent Finite-Element Methods for Singularly Perturbed Convection-Diffusion Equations. Habilitationsschrift, Mathematische Fakultät, Freiburg, 1998
7. K. Eriksson and C. Johnson. Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems. Math. Comput. 60, 167-188 (1993)
8. R. Eymard, T. Gallouet, M. Ghilani, and R. Herbin. Error estimates for the approximate solution of a nonlinear hyperbolic equation given by finite volume schemes. IMA Journal of Numerical Analysis 18, 563-594 (1998)
9. R. Helmig. Multiphase Flow and Transport Processes in the Subsurface: A contribution to the modeling of hydrosystems. Springer, Berlin, Heidelberg, 1997
10. R. Herbin. An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh. Numer.Meth.Part.Diff.Equ. 11, 165-173 (1995)
11. J. Jaffre. Decentrage et elements finis mixtes pour les equations de diffusion-convection. Calcolo 21, 171-197 (1984)
12. V. John, J.M. Maubach, and L. Tobiska. Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems. Numer. Math. 78(2), 165-188 (1997)
13. C. Johnson. Finite element methods for convection-diffusion problems. In Computing methods in applied sciences and engineering V, Proc. 5th int. Symp., Versailles 1981, pages 311-323, (1982)
14. D. Kröner. Numerical schemes for conservation laws. Teubner, Stuttgart, 1997
15. D. Kröner and M. Ohlberger. A-posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions. Math. Comput. 69, 25-39 (2000)
16. D. Kröner and M. Rokyta. A-priori error estimates for upwind finite volume schemes in several space dimensions. Preprint 37, Math. Fakultät, Freiburg, Dec. 1996
17. S.N. Kruzkov. First order quasilinear equations in several independent variables. Math. USSR Sbornik 10, 217-243 (1970)
18. N.N. Kuznetsov. Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation. USSR, Comput. Math. and Math. Phys. 16(6), 159-193 (1976)
19. J. Málek, J. Nečas, M. Rokyta, and M. Růžička. Weak and Measure-valued Solutions to Evolutionary PDEs., volume 13 of Applied Mathematics and Mathematical Computation. Chapman and Hall, London, Weinheim, New York, Tokyo, Melbourne, Madras, 1968
20. M. Ohlberger. Convergence of a Mixed Finite Element - Finite Volume Method for the Two Phase Flow in Porous Media. East-West J. Numer. Math. 5, 183-210 (1997)
21. H.-G. Roos, M. Stynes, and L. Tobiska. Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems., volume 24 of Springer Series in Computational Mathematics. Springer, Berlin, 1996
22. Tadmor, E. Local error estimates for discontinuous solutions of nonlinear hyperbolic equations. SIAM J. Numer. Anal. 28, 891-906 (1991)
23. R. Verfürth. A posteriori error estimators for convection-diffusion equations. Numer. Math. 80, 641-663 (1998)
24. J.P. Vila. Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I Explicit monotone schemes. RAIRO, Modelisation Math. Anal. Numer. 28, 267-295 (1994)