Convergence of a finite volume scheme for nonlinear degenerate parabolic equations (Q1614994)

The finite volume approximation scheme for the entropy weak solution \(u\) of the nonlinear degenerate parabolic equation \[ u_t+ \text{div}(qf(u))- \Delta\varphi(u)= 0 \] by a piecewise constant function \(u_{{\mathcal D}}\) using a discretization \({\mathcal D}\) and in time is proposed. If space and time steps tend to zero, the convergence of \(u_{{\mathcal D}}\) to \(u\) is proved. In the first step the estimates on \(u_{{\mathcal D}}\) are used to prove the convergence, up to a subsequences, of \(u_{{\mathcal D}}\) to a measure valued entropy solution, called an entropy process solution. A result of uniqueness of the entropy process solution is proved, yielding the strong convergence of \(u_{{\mathcal D}}\) to \(u\). Some numerical results concerning a model equation are presented.