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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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