Convergence of a finite volume scheme for nonlinear degenerate parabolic equations

Summary.

One approximates the entropy weak solution u of a nonlinear parabolic degenerate equation $u_t+{\rm div}({\mathbf q} f(u))-\Delta \phi(u)=0$ by a piecewise constant function $u_{{\mathcal D}}$ using a discretization ${\mathcal D}$ in space and time and a finite volume scheme. The convergence of $u_{{\mathcal D}}$ to u is shown as the size of the space and time steps tend to zero. In a first step, estimates on $u_{{\mathcal D}}$ are used to prove the convergence, up to a subsequence, of $u_{{\mathcal D}}$ 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 $u_{{\mathcal D}}$ to{\it u}. Some on a model equation are shown.