Convergence of a finite volume scheme for nonlinear degenerate parabolic equations

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.

Authors and Affiliations

1. Université de Marne-la-Vallée, Champs-sur-Marne, 77454 Marne-la-Vallée, France; e-mail: eymard@math.univ-mlv.fr , , , , , , FR

   Robert Eymard

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

   Thierry Gallouït, Raphaèle Herbin & Anthony Michel

Received September 27, 2000 / Published online October 17, 2001

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