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

Mario Ohlberger

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

  • Volume: 35, Issue: 2, page 355-387
  • ISSN: 0764-583X

Abstract

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This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation c t + ∇ · ( 𝐮 f ( c ) ) - ∇ · ( D ∇ c ) + λ c = 0 . The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L 1 -norm, independent of the diffusion parameter D . The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical experiments underline the applicability of the theoretical results.

How to cite

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Ohlberger, Mario. "A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.2 (2001): 355-387. <http://eudml.org/doc/194054>.

@article{Ohlberger2001,
abstract = {This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation $c_t + \nabla \cdot ( \mathbf \{u\}f(c)) - \nabla \cdot (D \nabla c) + \lambda c = 0$. The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the $L^1$-norm, independent of the diffusion parameter $D$. The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical experiments underline the applicability of the theoretical results.},
author = {Ohlberger, Mario},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {a posteriori error estimates; convection diffusion reaction equation; finite volume schemes; adaptive methods; unstructured grids; numerical experiments},
language = {eng},
number = {2},
pages = {355-387},
publisher = {EDP-Sciences},
title = {A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations},
url = {http://eudml.org/doc/194054},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Ohlberger, Mario
TI - A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 2
SP - 355
EP - 387
AB - This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation $c_t + \nabla \cdot ( \mathbf {u}f(c)) - \nabla \cdot (D \nabla c) + \lambda c = 0$. The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the $L^1$-norm, independent of the diffusion parameter $D$. The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical experiments underline the applicability of the theoretical results.
LA - eng
KW - a posteriori error estimates; convection diffusion reaction equation; finite volume schemes; adaptive methods; unstructured grids; numerical experiments
UR - http://eudml.org/doc/194054
ER -

References

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  1. [1] L. Angermann, An introduction to finite volume methods for linear elliptic equations of second order. Preprint 164, Institut für Angewandte Mathematik, Universität Erlangen (1995). MR1370105
  2. [2] Lutz Angermann, A finite element method for the numerical solution of convection-dominated anisotropic diffusion equations. Numer. Math. 85 (2000) 175–195. Zbl0956.65107
  3. [3] 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 (1998) 263–311. Zbl0942.76035
  4. [4] I. Babuška and W.C. Rheinboldt, Error estimators for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736–754. Zbl0398.65069
  5. [5] E. Bänsch, Local mesh refinement in 2 and 3 dimensions. IMPACT Comput. Sci. Engrg. 3 (1991) 181–191. Zbl0744.65074
  6. [6] 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 (1996) 237–264. Zbl0868.65076
  7. [7] F. Bouchut and B. Perthame, Kruzkov’s estimates for scalar conservation laws revisited. Trans. Amer. Math. Soc. 350 (1998) 2847–2870. Zbl0955.65069
  8. [8] J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147 (1999) 269–361. Zbl0935.35056
  9. [9] C. Chainais-Hillairet, Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimates. ESAIM: M2AN 33 (1999) 129–156. Zbl0921.65071
  10. [10] S. Champier, Error estimates for the approximate solution of a nonlinear hyperbolic equation with source term given by finite volume scheme. Preprint, UMR 5585, Saint-Étienne University (1998). Zbl1015.65043MR1901418
  11. [11] G. Chavent and J. Jaffre, Mathematical models and finite elements for reservoir simulation. Elsevier, New York (1986). Zbl0603.76101
  12. [12] B. Cockburn, F. Coquel and P.G. Lefloch, An error estimate for finite volume methods for multidimensional conservation laws. Math. Comput. 63 (1994) 77–103. Zbl0855.65103
  13. [13] B. Cockburn and H. Gau, A posteriori error estimates for general numerical methods for scalar conservation laws. Comput. Appl. Math. 14 (1995) 37–47. Zbl0834.65091
  14. [14] B. Cockburn and P.A. Gremaud, A priori error estimates for numerical methods for scalar conservation laws. Part I: The general approach. Math. Comput. 65 (1996) 533–573. Zbl0848.65067
  15. [15] B. Cockburn and G. Gripenberg, Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations. J. Differential Equations 151 (1999) 231–251. Zbl0921.35017
  16. [16] W. Dörfler, Uniformly convergent finite-element methods for singularly perturbed convection-diffusion equations. Habilitationsschrift, Mathematische Fakultät, Freiburg (1998). 
  17. [17] K. Eriksson and C. Johnson, Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems. Math. Comput. 60 (1993) 167–188. Zbl0795.65074
  18. [18] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. II: Optimal error estimates in L ∞ L 2 and L ∞ L ∞ . SIAM J. Numer. Anal. 32 (1995) 706–740. Zbl0830.65094
  19. [19] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. IV: Nonlinear Problems. SIAM J. Numer. Anal. 32 (1995) 1729–1749. Zbl0835.65116
  20. [20] S. Evje, K.H. Karlsen and N.H. Risebro, A continuous dependence result for nonlinear degenerate parabolic equations with spatial dependent flux function. Preprint, Department of Mathematics, Bergen University (2000). MR1882934
  21. [21] R. Eymard, T. Gallouët, M. Ghilani and R. Herbin, Error estimates for the approximate solution of a nonlinear hyperbolic equation given by finite volume schemes. IMA J. Numer. Anal. 18 (1998) 563–594. Zbl0973.65078
  22. [22] R. Eymard, T. Gallouët, R. Herbin and A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Preprint LATP 00-20, CMI, Provence University, Marseille (2000). Zbl1005.65099MR1917365
  23. [23] P. Frolkovic, Maximum principle and local mass balance for numerical solutions of transport equations coupled with variable density flow. Acta Math. Univ. Comenian. 67 (1998) 137–157. Zbl0940.76039
  24. [24] J. Fuhrmann and H. Langmach, Stability and existence of solutions of time-implicit finite volume schemes for viscous nonlinear conservation laws. Preprint 437, Weierstraß-Institut, Berlin (1998). Zbl0978.65081MR1825125
  25. [25] R. Helmig, Multiphase flow and transport processes in the subsurface: A contribution to the modeling of hydrosystems. Springer, Berlin, Heidelberg (1997). 
  26. [26] R. Herbin, An error estimate for a finite volume scheme for a diffusion-convection problem on a triangular mesh. Numer. Methods Partial Differential Equation 11 (1995) 165–173. Zbl0822.65085
  27. [27] P. Houston and E. Süli, Adaptive lagrange-galerkin methods for unsteady convection-dominated diffusion problems. Report 95/24, Numerical Analysis Group, Oxford University Computing Laboratory (1995). Zbl0957.65085
  28. [28] J. Jaffre, Décentrage et élements finis mixtes pour les équations de diffusion-convection. Calcolo 21 (1984) 171–197. Zbl0562.65077
  29. [29] V. John, J.M. Maubach and L. Tobiska, Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems. Numer. Math. 78 (1997) 165–188. Zbl0898.65068
  30. [30] C. Johnson, Finite element methods for convection-diffusion problems, in Proc. 5th Int. Symp. (Versailles, 1981), Computing methods in applied sciences and engineering V (1982) 311–323. Zbl0505.76099
  31. [31] K.H. Karlsen and N.H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Preprint 143, Department of Mathematics, Bergen University (2000). Zbl1027.35057MR1974417
  32. [32] D. Kröner, Numerical schemes for conservation laws. Teubner, Stuttgart (1997). Zbl0872.76001MR1437144
  33. [33] D. Kröner and M. Ohlberger, A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions. Math. Comput. 69 (2000) 25–39. Zbl0934.65102
  34. [34] D. Kröner and M. Rokyta, A priori error estimates for upwind finite volume schemes in several space dimensions. Preprint 37, Math. Fakultät, Freiburg (1996). Zbl0872.65093
  35. [35] S.N. Kruzkov, First order quasilinear equations in several independent variables. Math. USSR Sbornik 10 (1970) 217–243. Zbl0215.16203
  36. [36] N.N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation. USSR, Comput. Math. Math. Phys. 16 (1976) 159–193. Zbl0381.35015
  37. [37] J. Málek, J. Nečas, M. Rokyta and M. Růžička, Weak and measure-valued solutions to evolutionary PDEs, in Applied Mathematics and Mathematical Computation 13, Chapman and Hall, London, Weinheim, New York, Tokyo, Melbourne, Madras (1968). Zbl0851.35002MR1409366
  38. [38] M. Marion and A. Mollard, An adaptive multi–level method for convection diffusion problems. ESAIM: M2AN 34 (2000) 439–458. Zbl0952.65067
  39. [39] R.H. Nochetto, A. Schmidt and C. Verdi, A posteriori error estimation and adaptivity for degenerate parabolic problems. Math. Comput. 69 (2000) 1–24. Zbl0942.65111
  40. [40] 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 (1997) 183–210. Zbl0899.76261
  41. [41] M. Ohlberger, A posteriori error estimates for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations. Numer. Math. 87 (2001) 737–761. Zbl0973.65076
  42. [42] Ch. Rohde, Entropy solutions for weakly coupled hyperbolic systems in several space dimensions. Z. Angew. Math. Phys. 49 (1998) 470–499. Zbl0906.35058
  43. [43] Ch. Rohde, Upwind finite volume schemes for weakly coupled hyperbolic systems of conservation laws in 2D. Numer. Math. 81 (1998) 85–123. Zbl0918.35086
  44. [44] H.-G. Roos, M. Stynes and L. Tobiska, Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems, in Springer Ser. Comput. Math. 24, Springer-Verlag, Berlin (1996). Zbl1155.65087MR1477665
  45. [45] E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations. SIAM J. Numer. Anal. 28 (1991) 891–906. Zbl0732.65084
  46. [46] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques, in Wiley-Teubner Ser. Adv. Numer. Math., Teubner, Stuttgart (1996). Zbl0853.65108
  47. [47] R. Verfürth, A posteriori error estimators for convection-diffusion equations. Numer. Math. 80 (1998) 641–663. Zbl0913.65095
  48. [48] J.P. Vila, Convergence and error estimates in finite volume schemes for general multi-dimensional scalar conservation laws. I Explicit monotone schemes. ESAIM: M2AN 28 (1994) 267–295. Zbl0823.65087

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