Adaptive finite volume approximations for weakly coupled convection dominated parabolic systems

IMA Journal of Numerical Analysis

Volumn 22, Issue 2, 2002, Pages 253-280

  Ohlberger, Mario ^a   Rohde, Christian ^a  

^a UNIVERSITY OF FREIBURG   (Germany)

Author keywords

A posteriori error estimates; Adaptive finite volume methods; Convection diffusion reaction equation; Unstructured grids; Weakly coupled system

Indexed keywords

ADAPTIVE ALGORITHMS; DIFFUSION IN LIQUIDS; FINITE VOLUME METHOD; HEAT CONVECTION;

A-POSTERIORI ERROR ESTIMATES; ADAPTIVE FINITE VOLUME METHOD; CONVECTION-DIFFUSION-REACTION EQUATIONS; UNSTRUCTURED GRID; WEAKLY COUPLED SYSTEMS;

ERROR ANALYSIS;

EID: 0036243751     PISSN: 02724979     EISSN: None     Source Type: Journal    
DOI: 10.1093/imanum/22.2.253     Document Type: Article

References (34)

1. ANGERMANN, L. (1995) An introduction to finite volume methods for linear elliptic equations of second order. Preprint 164. Institut für Angewandte Mathematik, Universität Erlangen.
2. ANGOT, P., DOLEJŠÍ, V., FEISTAUER, M. & FELCMAN, J. (1998) Analysis of a combined barycentric finite volume-finite element method for nonlinear convection diffusion problems. Appl. Math. Praha, 43, 263-311.
3. CARRILLO, J. (1999) Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal., 147, 269-361.
4. CHAINAIS-HILLAIRET, C. (1999) Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimates. M2AN Math. Model. Numer. Anal., 33, 129-156.
5. CHUCK, K. N., CONLY, C. C. & SMOLLER, J. A. (1997) Positively invarient regions for systems of nonlinear diffusion equations. Indiana Univ. Math., 26, 373-392.
6. CIRPKA, O., FRIND, E. & HELMIG, R. (1999) Numerical simulation of biodegradation controlled by transverse mixing. J. Contam. Hydrol., 40, 159-182.
7. COCKBURN, B. & GAU, H. (1995) A posteriori error estimates for general numerical methods for scalar conservation laws. Comput. Appl. Math., 14, 37-47.
8. ERIKSSON, K. & JOHNSON, C. (1993) Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems. Math. Comput., 60, 167-188.
9. EYMARD, R., GALLOUËT, T., GHILANI, M. & HERBIN, R. (1998) Error estimates for the approximate solution of a nonlinear hyperbolic equation given by finite volume schemes. IMA J. Numer. Anal., 18, 563-594.
10. EYMARD, R., GALLOUËT, T. & HERBIN, R. (2000a) Error estimates for approximate solutions of a nonlinear convection-diffusion problem. Preprint. CMI, Université de Provence, Marseille.
11. EYMARD, R., GALLOUËT, T., HERBIN, R. & MICHEL, A. (2000b) Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Preprint LATP 00-20, CMI, Universitë de Provence, Marseille. To appear in Numer. Math.
12. FREISTÜHLER, H. & ROHDE, C. (2001) Numerical computation of viscous profiles for hyperbolic conservation laws. Math. Comput. To appear.
13. FROLKOVIC, P. (1998) Maximum principle and local mass balance for numerical solutions of transport equations coupled with variable density flow. Acta Math. Univ. Comen., 67, 137-157.
14. FROLKOVIC, P. & GEISER, J. (2000) Numerical simulations of radionuclides transport in double porosity media with sorption. Proceedings of Algorithmy 2000. pp. 28-36.
15. FUHRMANN, J. & LANGMACH, H. (2001) Stability and existence of solutions of time-implicit finite Volume schemes for viscous nonlinear conservation laws. Appl. Numer. Math., 37, 201-230.
16. GEISER, J. (2001) Numerical simulation of a model for transport and reaction of radionuclides. Proceedings of LSSC 01. Submitted.
17. HERBIN, R. (1995) 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.
18. HOLDEN, H., KARLSEN, K. H. & RISEBRO, N.-H. (2002) On the uniqueness and existence of entropy solutions of weakly coupled systems of nonlinear degenerate parabolic equations. Preprint, Department of Mathematics, Universtiy of Bergen.
19. HOUSTON, P. & SÜLI, E. (2001) Adaptive Lagrange-Galerkin methods for unsteady convection-diffusion problems. Math. Comput., 70, 77-106.
20. JAKOBSEN, E. R. & KARLSEN, K. H. (2001) Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations. Technical Report, UCLA Computational and Applied Mathematics Report. To appear in J. Diff. Equ.
21. JOHN, V., MAUBACH, J. M. & TOBISKA, L. (1997) Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems. Numer. Math., 78, 165-188.
22. JOHNSON, C. (1982) Finite element methods for convection-diffusion problems. Computing Methods in Applied Sciences and Engineering V (Proc. 5th Int. Symp., Versailles 1981). pp. 311-323.
23. KARLSEN, K. H. & RISEBRO, N. H. (2000) On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Preprint 143. Department of Mathematics, University of Bergen.
24. KRÖNER, D. & OHLBERGER, M. (2000) A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions. Math. Comput., 69, 25-39.
25. KRÖNER, D. & ROKYTA, M. (1996) A priori error estimates for upwind finite volume schemes in several space dimensions. Preprint 37. Math. Fakultät, Freiburg.
26. KRUZKOV, S. N. (1970) First order quasilinear equations in several independent variables. Math. USSR Sbornik, 10, 217-243.
27. KUZNETSOV, N. N. (1976) Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation. USSR Comput. Math. Math. Phys., 16, 159-193.
28. MAJDA, A. (1981) A qualitative model for dynamic combustion. SIAM J. Appl. Math., 41, 70-93.
29. NORBURY, J. & STUART, A. M. (1989) A model for porous-medium combustion. Q. J. Mech. Appl. Math., 42, 159-178.
30. OHLBERGER, M. (2001a) A posteriori error estimates for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations. Numer. Math., 87, 737-761.
31. OHLBERGER, M. (2001b) A posteriori error estimates for vertex centred finite volume approximations of convection-diffusion-reaction equations. M2AN Math. Model. Numer. Anal., 35, 355-387.
32. ROHDE, C. (1998) Entropy solutions for weakly coupled hyperbolic systems in several space dimensions. Z. Angew. Math. Phys., 49, 470-499.
33. TADMOR, E. (1991) Local error estimates for discontinuous solutions of nonlinear hyperbolic equations. SIAM J. Numer. Anal., 28, 891-906.
34. VERFÜRTH, R. (1998) A posteriori error estimators for convection-diffusion equations. Numer. Math., 80, 641-663.