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Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. (English) Zbl 1005.65099
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.

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
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