Convergence of Finite Volume Approximations for a Nonlinear Elliptic-Parabolic Problem: A "Continuous" Approach

We study the approximation by finite volume methods of the model parabolicelliptic problem$b(v)_{t} = div (\vert Dv\vert^{p-2}Dv)$on$(0,T) \times \Omega \subset \mathbb{R} \times \mathbb{R}^d$with an initial condition and the homogeneous Dirichlet boundary condition. Because of the nonlinearity in the elliptic term, a careful choice of the gradient approximation is needed. We prove the convergence of discrete solutions to the solution of the continuous problem as the discretization step h tends to 0, under the main hypotheses that the approximation of the operator div$(\vert Dv\vert^{p-2}Dv)$provided by the finite volume scheme is still monotone and coercive, and that the gradient approximation is exact on the affine functions of$x \in \Omega$. An example of such a scheme is given for a class of two-dimensional meshes dual to triangular meshes, in particular for structured rectangular and hexagonal meshes. The proof uses the rewriting of the discrete problem under a "continuous" form. This permits us to directly apply the Alt-Luckhaus variational techniques which are known for the continuous case.