ANDREIANOV, Boris; BENDAHMANE, Mostafa; KARLSEN, Kenneth Hvistendahl

doi:10.1142/S0219891610002062

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ANDREIANOV, Boris
Laboratoire de Mathématiques de Besançon (UMR 6623) [LMB]

BENDAHMANE, Mostafa
Centro de Investigación en Ingeniería Matemática [Concepción] [CI²MA]
Institut de Mathématiques de Bordeaux [IMB]

KARLSEN, Kenneth Hvistendahl
Center of Mathematics for Applications [Oslo] [CMA]

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Ce document a été publié dans

Journal of Hyperbolic Differential Equations. 2010, vol. 7, n° 1, p. 1--67

World Scientific Publishing

Résumé en anglais

We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omn`es [43]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around basic a priori estimates, the discrete duality features, Minty–Browder type arguments, and “hyperbolic” L∞ weak-* compactness arguments (i.e. propagation of compactness along the lines of Tartar, DiPerna, . . . ). Our results cover the case of non-Lipschitz nonlinearities.< Réduire

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convergence

Mots clés en anglais

Degenerate hyperbolic-parabolic equation

conservation law

Leray–Lions type operator

non-Lipschitz flux

entropy solution

existence

uniqueness

finite volume scheme

discrete duality

convergence.

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Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations

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