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hal-00608521, version 1

## Convergence of approximate solutions to an elliptic-parabolic equation without the structure condition

Boris Andreianov () 1, Petra Wittbold 2

(2011-07-13)

Abstract: We study the Cauchy-Dirichlet problem for the elliptic-parabolic equation $$b(u)_t +\div F(u) - \Delta u=f$$ in a bounded domain. We do not assume the structure condition ''$b(z)=b(\hat z) \Rightarrow F(z)=F(\hat z)$''. Our main goal is to investigate the problem of continuous dependence of the solutions on the data of the problem and the question of convergence of discretization methods. As in the work of Ammar and Wittbold \cite{AmmarWittbold} where existence was established, monotonicity and penalization are the main tools of our study. In the case of a Lipschitz continuous flux $F$, we justify the uniqueness of $u$ (the uniqueness of $b(u)$ is well-known) and prove the continuous dependence in $L^1$ for the case of strongly convergent finite energy data. We also prove convergence of the $\varepsilon$-discretized solutions used in the semigroup approach to the problem; and we prove convergence of a monotone time-implicit finite volume scheme. In the case of a merely continuous flux $F$, we show that the problem admits a maximal and a minimal solution.

• 1:  Laboratoire de Mathématiques (LM-Besançon)
• CNRS : UMR6623 – Université de Franche-Comté
• 2:  Essen, Fakultaet fuer Mathematik
• Universität Duisburg-Essen
• Collaboration : Procope project No.21806TL
• Domain : Mathematics/Analysis of PDEs
Mathematics/Numerical Analysis
• Keywords : Elliptic-Parabolic Equation – Structure Condition – Convergence of Approximate Solutions – Finite Volume Method

• hal-00608521, version 1
• oai:hal.archives-ouvertes.fr:hal-00608521
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• Submitted on: Wednesday, 13 July 2011 12:28:35
• Updated on: Wednesday, 13 July 2011 15:16:18