Error estimates for a mixed finite element discretization of some degenerate parabolic equations

  • Authors:

  • Florin A. Radu;Iuliu Sorin Pop;Peter Knabner

  • Affiliations:

  • Max-Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103, Leipzig, Germany and UFZ, Helmholtz Center for Environmental Research, Permoserstr. 15, 04318, Leipzig, Germany;Eindhoven University of Technology, Department of Mathematics and Computer Science, P.O. Box 513, Inselstr. 22, 5600, MB Eindhoven, The Netherlands;University Erlangen-Nürnberg, Institute of Applied Mathematics, P.O. Box 513, Martensstr. 3, 91058, Erlangen, Germany

  • Venue:
  • Numerische Mathematik
  • Year:
  • 2008

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Abstract

We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases. A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based on the mixed finite element method (MFEM) in space, and is of one step implicit in time. The lowest order Raviart–Thomas elements are used. Here we extend the results in Radu et al. (SIAM J Numer Anal 42:1452–1478, 2004), Schneid et al. (Numer Math 98:353–370, 2004) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart–Thomas elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples.