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DISCRETE DUALITY FINITE VOLUME SCHEMES FOR DOUBLY NONLINEAR DEGENERATE HYPERBOLIC-PARABOLIC EQUATIONS

Laboratoire de Mathématiques, Université de Franche-Comté, 16 route de Gray, 25 030 Besançon, Cedex, France

Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C Concepción, Chile

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, and

K. H. KARLSEN

Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway

Center for Biomedical Computing, Simula Research Laboratory, P.O. Box 134, N–1325 Lysaker, Norway

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https://doi.org/10.1142/S0219891610002062Cited by:41 (Source: Crossref)

Abstract

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 Omnès [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.

Keywords:

AMSC: Primary 35K65, Primary 35L65, Primary 74S10, Secondary 65M12, Secondary 35A02, Secondary 35A01

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