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Vietnam Journal of Mathematics

, Volume 45, Issue 1–2, pp 179–198 | Cite as

A Priori Error Analysis for the Galerkin Finite Element Semi-discretization of a Parabolic System with Non-Lipschitzian Nonlinearity

  • Peter Knabner
  • Rolf Rannacher
Article
  • 96 Downloads

Abstract

This paper deals with the numerical approximation of certain degenerate parabolic systems arising from flow problems in porous media with slow adsorption. The characteristic difficulty of these problems comes from their monotone but non-Lipschitzian nonlinearity. For a model problem of this type, optimal-order pointwise error estimates are derived for the spatial semi-discretization by the finite element Galerkin method. The proof is based on linearization through a parabolic duality argument in L (L ) spaces and corresponding sharp L 1 estimates for regularized parabolic Green functions.

Keywords

Degenerate parabolic problem Non-Lipschitzian nonlinearity Finite element method Pointwise error Porous media flow 

Mathematics Subject Classification (2010)

65M15 35K65 35R35 76S05 

References

  1. 1.
    Barrett, J., Knabner, P.: Finite element approximation of transport of reactive solutes in porous media, Part 1. Error estimates for non-equilibrium adsorption processes. SIAM J. Numer. Anal. 34, 201–227 (1997)CrossRefzbMATHGoogle Scholar
  2. 2.
    Barrett, J., Knabner, P.: Finite element approximation of transport of reactive solutes in porous media, Part 2. Error estimates for equilibrium adsorption processes. SIAM J. Numer. Anal. 34, 455–479 (1997)CrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, H.: An L 2- and L -error analysis for parabolic finite element equations with application to superconvergence and error expansion. Dissertation, University of Heidelberg (1993)Google Scholar
  4. 4.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, vol 40. SIAM, Philadelphia (2002)CrossRefGoogle Scholar
  5. 5.
    deal.II: A Finite Element Differential Equations Analysis Library. Release 7.0, www.dealii.org (2011)
  6. 6.
    Douglas, J., Dupont, T., Wahlbin, L.: The stability in L q of the L 2-projection into finite element function spaces. Numer. Math. 23, 193–197 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    van Duijn, C.J., Knabner, P.: Solute transport through porous media with slow adsorption. In: Hoffmann, K.H., Sprekels, J. (eds.) Free Boundary Problems: Theory and Applications, Vol. I, pp 375–388. Longman, White Plains (1990)Google Scholar
  8. 8.
    van Duijn, C.J., Knabner, P.: Solute transport in porous media with equilibrium and non-equilibrium multiple-site adsorption: traveling waves. J. Reine Angew. Math. 415, 1–49 (1991)zbMATHGoogle Scholar
  9. 9.
    Johnson, C., Larsson, S., Thomée, V., Wahlbin, L.: Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data. Math. Comput. 49, 331–357 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Knabner, P.: Mathematische Modelle für Transport und Sorption gelöster Stoffe in porösen Medien. Verlag P. Lang, Frankfurt a.M. (1991)zbMATHGoogle Scholar
  11. 11.
    Lambrecht, M.: Finite-Elemente-Approximation eines Diffusionsproblems mit nicht-Lipschitz-stetiger Nichtlinearität. Diploma thesis, Heidelberg University (2011)Google Scholar
  12. 12.
    Luskin, M., Rannacher, R.: On the smoothing property of the Galerkin method for parabolic equations. SIAM J. Numer. Anal. 19, 93–113 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nochetto, R.H.: Sharp L -error estimates for semilinear elliptic problems with free boundaries. Numer. Math. 54, 243–255 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Nochetto, R.H., Verdi, C.: Approximation of degenerate parabolic problems using numerical integration. SIAM J. Numer. Anal. 25, 784–814 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Rannacher, R.: L -stability estimates and asymptotic error expansions for parabolic finite element equations. In: Frehse, J., Rannacher, R. (eds.) Proceedings of GAMM-Seminar “Extrapolation and Defect Correction Methods”. University of Heidelberg, June 22–23, 1990. Bonn. Math. Schr., vol. 228, pp. 74–94 (1991)Google Scholar
  16. 16.
    Rannacher, R.: Pointwise convergence of finite element approximations on irregular meshes. Preprint, University of Heidelberg (1993)Google Scholar
  17. 17.
    Schatz, A., Thomée, V., Wahlbin, L.: Maximum norm stability and error estimates in parabolic finite element equations. Commun. Pure Appl. Math. 33, 265–304 (1980)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Lehrstuhl für Angewandte Mathematik IUniversität Erlangen-Nürnberg, Department MathematikErlangenGermany
  2. 2.Institut für Angewandte MathematikUniversität HeidelbergHeidelbergGermany

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