  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 \cite{DomOmnes}) 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 some basic a priori estimates, the discrete duality features,
Minty-Browder type arguments, and "hyperbolic" $L^\infty$ weak-$\star$
compactness arguments (i.e., propagation of compactness along the lines of
Tartar, DiPerna, ...). Our results cover the case of non-Lipschitz
nonlinearities.
