A posteriori error estimates for finite volume element approximations of convection-diffusion-reaction equations

Computational Geosciences

Volumn 6, Issue 3-4, 2002, Pages 483-503

  Lazarov, Raytcho ^a   Tomov, Stanimire ^a  

^a TEXAS A AND M UNIVERSITY   (United States)

Author keywords

Adaptive grid refinement; Convection diffusion; Finite volume method; Residual error estimators; Up wind approximation

Indexed keywords

DIFFUSION-CONVECTION THEORY; FINITE VOLUME TECHNIQUE; FLUID FLOW; MATHEMATICAL METHOD; POROUS MEDIUM; REACTION-DIFFUSION EQUATION;

CONVECTION; DIFFUSION; FINITE VOLUME METHOD; FLUID FLOW; NUMERICAL METHOD; POROUS MEDIUM; REACTIVE TRANSPORT;

EID: 0036930181     PISSN: 14200597     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1021247300362     Document Type: Article

References (32)

1. L. Angermann, An a-posteriori estimation for the solution of an elliptic singularly perturbed problem, IMA J. Numer. Anal. 12 (1992) 201-215.
2. L. Angermann, Balanced a posteriori error estimates for finite-volume type discretizations of convection-dominated elliptic problems, Computing 55(4) (1995) 305-324.
3. D.N. Arnold, A. Mukherjee and L. Pouly, Locally adapted tetrahedral meshes using bisection, SIAM J. Sci. Comput. 22(2) (2000) 431-448.
4. I. Babuska and W.C. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736-754.
5. R.E. Bank and R.K. Smith, A posterior error estimates based on hierarchical bases, SIAM J. Numer. Anal. 30 (1993) 921-932.
6. R. Becker, C. Johnson and R. Rannacher, Adaptive error control for multigrid finite element methods, Computing 55(4) (1995) 271-288.
7. R. Becker and R. Rannacher, A feed-back approach to error control in finite element methods: Basic analysis and examples, East-West J. Numer. Math. 4(4) (1996) 237-264.
8. 1 (Ω), Math. Comp. 71(237) (2002) 147-156.
9. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, Berlin, 1996).
10. Z. Cai, On the finite volume element method, Numer. Math. 58 (1991) 713-735.
11. Z. Cai, J. Mandel and S. McCormick, The finite volume element method for diffusion equations on general triangulations, SIAM J. Numer. Anal. 38 (1991) 392-402.
12. C. Carstensen, Quasi-interpolation and a posteriori error analysis in finite element methods, Math. Model. Numer. Anal. 33 (1999) 1187-1202.
13. P. Clément, Approximation by finite element functions using local regularization, RAIRO Anal. Numér. 9 (1975) 77-85.
14. K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Computational Differential Equations (Cambridge Univ. Press, Cambridge, 1996).
15. K. Eriksson and C. Johnson, An adaptive finite element method for linear elliptic problems, Math. Comp. 50 (1988) 361-382.
16. R.E. Ewing, R.D. Lazarov and Y. Lin, Finite volume element approximations for non-local reactive flows in porous media, Numer. Methods Partial Differential Equations 16 (2000) 285-311.
17. T. Ikeda, Maximum principle in finite element models for convection-diffusion phenomena, in: Lecture Notes in Numerical Applied Analysis, Vol.4 (North-Holland, Amsterdam, 1983).
18. C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method (Cambridge Univ. Press, Cambridge, 1995).
19. R.D. Lazarov, J.E. Pasciak and S.Z. Tomov, Error control, local grid refinement and efficient solution algorithms for singularly perturbed problems, in: Analytical and Numerical Methods for Convection-Dominated and Singularly Perturbed Problems, eds. L.G. Vulkov, J.J.H. Miller and G.I. Shishkin (NOVA Science, 2000) pp. 71-82.
20. R.H. Li and Z.Y. Chen, The Generalized Difference Method for Differential Equations (Jilin Univ. Publ. House, 1994).
21. I.D. Mishev, Finite volume methods on Voronoi meshes, Numer. Methods Partial Differential Equations 14 (1998) 193-212.
22. R. Rodriguez, Some remarks on Zienkiewicz-Zhu estimator, Numer. Methods Partial Differential Equations 10 (1994) 625-635.
23. H.-O. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations (Springer, Berlin, 1996).
24. A.A. Samarskii, The Theory of Difference Schemes (Marcel Dekker, New York, 2001).
25. M. Slodicka and R. Van Keer, A nonlinear elliptic equation with non-local boundary condition solved by linearization, Preprint 1, Department of Mathematics, University of Gent, Belgium (2000).
26. M. Tabata, A finite element approximation corresponding to the up-wind finite differencing, Mem. Numer. Math. 4 (1977) 47-63.
27. S.Z. Tomov, Tool-box for parallel adaptive computations of 3-D convection-diffusion problems using domain decomposition, Technical Report ISC-00-10-MATH, Texas A&M University (2000).
28. R. Verfürth, A Review of a Posteriori Error Estimators and Adaptive Mesh Refinement Techniques (Teubner/Wiley, Stuttgart, 1996).
29. R. Verfürth, A posteriori error estimation and adaptive mesh-refinement techniques, J. Comput. Appl. Math. 50 (1994) 67-83.
30. J. Xu and L. Zikatanov, A monotone finite element scheme for convection-diffusion equations, Math. Comp. 68 (1999) 1429-1447.
31. O.C. Zienkiewicz and J.Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg. 24 (1987) 337-357.
32. O.C. Zienkiewicz and J.Z. Zhu, Adaptivity and mesh generation, Internat. J. Numer. Methods Engrg. 32 (1991).