Vol. LXXVI, 1 (2007)
p. 3 - 10

Finite volume schemes for nonlinear parabolic problems: another regularization method
R. Eymard, T. Gallouet and R. Herbin

Abstract.  On one hand, the existence of a solution to degenerate parabolic equations, without a nonlinear convection term, can be proven using the results of Alt and Luckhaus, Minty and Kolmogorov. On the other hand, the proof of uniqueness of an entropy weak solution to a nonlinear scalar hyperbolic equation, first provided by Krushkov, has been extended in two directions: Carrillo has handled the case of degenerate parabolic equations including a nonlinear convection term, whereas Di Perna has proven the uniqueness of weaker solutions, namely Young measure entropy solutions. All of these results are reviewed in the course of a convergence result for two regularizations of a degenerate parabolic problem including a nonlinear convective term. The first regularization is classicaly obtained by adding a minimal diffusion, the second one is given by a finite volume scheme on unstructured meshes. The convergence result is therefore only based on L¥ (W´(0,T)) and L2(0,T; H1(W)) estimates, associated with the uniqueness result for a weaker sense for a solution.

Keywords.  Degenerate parabolic equation, entropy weak solution, doubling variable technique, Young measures, finite volume scheme

AMS Subject classification.  Primary: 35K65, 35L60, 65M60.

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