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A Posteriori Error Estimates and Adaptive Methods for Hyperbolic and Convection Dominated Parabolic Conservation Laws

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Abstract

In this contribution we will give a survey on rigorous a posteriori error estimates and adaptive methods for finite volume approximations of hyperbolic and convection dominated parabolic conservation laws. Scalar problems are considered as well as weakly coupled systems where the coupling is only due to lower order terms. In the context of scalar hyperbolic conservation laws error estimates are obtained for cell centered finite volume schemes and for the staggered Lax—Friedrichs scheme in multi dimensions. In the case of weakly coupled convection dominated parabolic equations we get a posteriori error estimates for a vertex centered finite volume scheme which are uniform in the lower bound of the diffusion. Numerical experiments underline the applicability of the theoretical results in adaptive computations.

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References

  1. L. Angermann, P. Knabner, and K. Thiele. An error estimate for a finite volume discretization of density driven flow in porous media. Appl. Numer. Math., 26: 179–191, 1998.

    MathSciNet  MATH  Google Scholar 

  2. P. Arminjon, A. Madrane, and A. St-Cyr. Numerical simulation of 3-d flows with a non-oscillatory central scheme on unstructured tetrahedral grids. Talk presented at the eight international conference on hyperbolic problems, Magdeburg, 2000.

    Google Scholar 

  3. P. Arminjon, D. Stanescu, and M.C. Viallon. A two-dimensional finite volume extension of the Lax-Friedrichs and Nessyahu-Tadmor schemes for compressible flows. In 6th Int. Symposium on Comp. Fluid Dynamics, volume 4, pages 7–14, Lake Tahoe, Septempber 4–8, 1995.

    Google Scholar 

  4. Becker, R. and Rannacher, R. A feed-back approach to error control in finite element methods: Basic analysis and examples. East-West J. Numer. Math., 4: 237–264, 1996.

    MathSciNet  MATH  Google Scholar 

  5. D. Braess and R. Verfürth. A posteriori error estimators for the RaviartThomas element. SIAM J. Numer. Anal, 33: 2431–2444, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  6. C. Chainais-Hillairet. Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimates. M2AN Math. Model. Numer. Anal., 33: 129–156, 1999.

    Google Scholar 

  7. Cirpka, E. Frind, and R. Helmig. Numerical simulation of biodegradation controlled by transverse mixing. Journal of Contaminant Hydrology, 40: 159182, 1999.

    Google Scholar 

  8. 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.

    MathSciNet  MATH  Google Scholar 

  9. W.Dörfler. A convergent adaptive algorithm for poisson’s equation. SIAM J. Num. Anal, 33:1106–1124, 1996

    Google Scholar 

  10. W. Dörfler. Uniformly Convergent Finite—Element Methods for Singularly Perturbed Convection—Diffusion Equations. Habilitationsschrift, Mathematische Fakultät, Freiburg, 1998

    Google Scholar 

  11. K. Eriksson and C. Johnson. Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems. Math. Comput., 60: 167–188, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  12. K. Eriksson and C. Johnson. Adaptive finite element methods for parabolic problems. V: Long-time integration. SIAM J. Numer. Anal., 32: 1750–1763, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Feistauer, J. Felcman, M. Lukâcovâ-Medvid’ovâ, and G. Warnecke. Error estimates for a combined finite volume-finite element method for nonlinear convection-diffusion problems. SIAM J. Numer. Anal, 36(5):1528–1548, 1999

    Google Scholar 

  14. P.Houston, J.A. Mackenzie, E. Süli, and G. Warnecke. A posteriori error analysis for numerical approximation of Friedrichs system. Numer. Math, 82(3):433–470, 1999

    Google Scholar 

  15. 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.

    Google Scholar 

  16. W. Jäger and J. Kaeur. Solution of porous medium type systems by linear approximation schemes. Numer. Math, 60: 407–427, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  17. C. Johnson and A. Szepessy. Adaptive finite element methods for conservation laws based on a posteriori error estimates. Commun. Pure Appl. Math, 48: 199234, 1995.

    Google Scholar 

  18. Claes Johnson, Rolf Rannacher, and Mats Boman Numerics and hydrodynamic stability: Toward error control in computational fluid dynamics. SIAM J. Numer. Anal, 32 (4): 1058–1079, 1995.

    Google Scholar 

  19. 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.

    MATH  Google Scholar 

  20. M. Küther. Error estimates for the staggered Lax-Friedrichs scheme on unstructured grids. To appear in SIAM J. Numer. Anal, 2001.

    Google Scholar 

  21. 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.

    Google Scholar 

  22. P. D. Lax. Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math, 7: 159–193, 1954.

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Nessyahu and E. Tadmor. Non-oscillatory central differencing for hyperbolic conservations laws. J. Comput. Phys, 87: 408–463, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  24. M. Ohlberger. A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations. M2AN Math. Model. Numer. Anal, 35 (2): 355–387, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Ohlberger and C. Rohde. Adaptive finite volume approximations for weakly coupled convection dominated parabolic systems. Preprint, Mathematische Fakultät, Freiburg, April 2001.

    Google Scholar 

  26. C. Rohde. Entropy solutions for weakly coupled hyperbolic systems in several space dimensions. Z. Angew. Math. Phys, 49 (3): 470–499, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  27. Tadmor, E. Local error estimates for discontinuous solutions of nonlinear hyperbolic equations. SIAM J. Numer. Anal, 28: 891–906, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  28. R. Verfürth. Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation. Numer. Math, 78: 479–493, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  29. J.P. Vila. Convergence and error estimates in finite volume schemes for general multi-dimensional scalar conservation laws. I Explicit monotone schemes. RAIRO, Modelisation Math. Anal. Numer, 28: 267–295, 1994.

    Google Scholar 

  30. G.B. Whitham. Linear and nonlinear waves John Wiley * Sons Inc., New York, 1999. Reprint of the 1974 original, A Wiley-Interscience Publication.

    Google Scholar 

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Freiburg, Hermann-Herder-Straße 10, 79104, Freiburg, Germany

    Dietmar Kröner, Marc Küther, Mario Ohlberger & Christian Rohde

Authors
  1. Dietmar Kröner
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  2. Marc Küther
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  3. Mario Ohlberger
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  4. Christian Rohde
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Editor information

Editors and Affiliations

  1. Mathematics Department and Center for Scientific Computing, University of Warwick, CV4 7AL, Coventry, UK

    Markus Kirkilionis

  2. Institute of Scientific Computing, University of Heidelberg, Im Neuheimer Feld 368, 69120, Heidelberg, Germany

    Susanne Krömker

  3. Institute of Applied Mathematics, University of Heidelberg, Im Neuheimer Feld 293, 69120, Heidelberg, Germany

    Rolf Rannacher

  4. Department of Mathematics, University of Heidelberg, Im Neuheimer Feld 288, 69120, Heidelberg, Germany

    Friedrich Tomi

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© 2003 Springer-Verlag Berlin Heidelberg

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Kröner, D., Küther, M., Ohlberger, M., Rohde, C. (2003). A Posteriori Error Estimates and Adaptive Methods for Hyperbolic and Convection Dominated Parabolic Conservation Laws. In: Kirkilionis, M., Krömker, S., Rannacher, R., Tomi, F. (eds) Trends in Nonlinear Analysis. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05281-5_7

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  • DOI: https://doi.org/10.1007/978-3-662-05281-5_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-07916-0

  • Online ISBN: 978-3-662-05281-5

  • eBook Packages: Springer Book Archive

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