Convergence of a finite volume scheme for nonlinear degenerate parabolic equations

One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation u(t) +div(qf(u)) - Deltarho(u) = 0 by a piecewise constant function u(D) using a discretization D in space and time and a finite volume scheme. The convergence of u(D) to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on u(D) are used to prove the convergence, up to a subsequence, of u(D) to a measure valued entropy solution (called here an entropy process solution). A result of uniqueness of the entropy process solution is proved, yielding the strong convergence of u(D) to u. Some numerical results on a model equation are shown.