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Numerische Mathematik

, Volume 92, Issue 1, pp 41–82 | Cite as

Convergence of a finite volume scheme for nonlinear degenerate parabolic equations

  • Robert Eymard
  • Thierry Gallouït
  • Raphaèle Herbin
  • Anthony Michel
Original article

Summary.

One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation \(u_t+{\rm div}({\mathbf q} f(u))-\Delta \phi(u)=0\) by a piecewise constant function \(u_{{\mathcal D}}\) using a discretization \({\mathcal D}\) in space and time and a finite volume scheme. The convergence of \(u_{{\mathcal D}}\) to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on \(u_{{\mathcal D}}\) are used to prove the convergence, up to a subsequence, of \(u_{{\mathcal D}}\) to a measure valued entropy solution (called here an entropy process solution). A result of uniqueness of the entropy process solution is proved, yielding the strong convergence of \(u_{{\mathcal D}}\) to{\it u}. Some on a model equation are shown.

Mathematics Subject Classification: 65M12 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Robert Eymard
    • 1
  • Thierry Gallouït
    • 2
  • Raphaèle Herbin
    • 2
  • Anthony Michel
    • 2
  1. 1.Université de Marne-la-Vallée, Champs-sur-Marne, 77454 Marne-la-Vallée, France; e-mail: eymard@math.univ-mlv.fr FR
  2. 2.Université de Provence, Marseille, Centre de Math. et d'Informatique, Rue Joliot-Curie 39, 13453 Marseille, France; e-mail: {gallouet,herbin,michel}@cmi.univ-mrs.fr FR

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