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 parabolic-elliptic problem $b(v)_t={{\rm div \,}} (|Dv|^{p-2} Dv)$ on $(0,T)\times\Omega\subset \R\times\R^d$ 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 ${{\rm div \,}} (|Dv|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 $x\in \Om$. 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.