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A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems

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Abstract

We propose and analyze a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the diffusion term, which generally involves an inhomogeneous and anisotropic diffusion tensor, over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming (Crouzeix–Raviart) finite element method, or using the stiffness matrix of the hybridization of the lowest-order Raviart–Thomas mixed finite element method. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. Checking the local Péclet number, we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection-dominated case. This technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor. We prove the convergence of the scheme, only supposing the shape regularity condition for the original mesh. We use a priori estimates and the Kolmogorov relative compactness theorem for this purpose. The proposed scheme is robust, only 5-point (7-point in space dimension three), locally conservative, efficient, and stable, which is confirmed by numerical experiments.

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Authors and Affiliations

  1. Département de Mathématiques, Université de Marne-la-Vallée, 5 boulevard Descartes, Champs-sur-Marne, 77 454, Marne-la-Vallée, France

    Robert Eymard

  2. Laboratoire de Mathématiques, Analyse Numérique et EDP, Université de Paris-Sud et CNRS, Bât. 425, 91 405, Orsay, France

    Danielle Hilhorst & Martin Vohralík

  3. Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 120 00, Prague 2, Czech Republic

    Martin Vohralík

Authors
  1. Robert Eymard
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  2. Danielle Hilhorst
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  3. Martin Vohralík
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Correspondence to Martin Vohralík.

Additional information

This work was supported by the GdR MoMaS, CNRS-2439, ANDRA, BRGM, CEA, EdF, France.

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Eymard, R., Hilhorst, D. & Vohralík, M. A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems. Numer. Math. 105, 73–131 (2006). https://doi.org/10.1007/s00211-006-0036-z

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