Convergence of approximate solutions to an elliptic-parabolic equation without the structure condition
type: Item
instance of: scholarly article
title: Convergence of approximate solutions to an elliptic-parabolic equation without the structure condition
zbMATH Open document ID: 1261.35094
author: Boris Andreianov
author: Petra Wittbold
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2013-01-04T00:00:00Z
review text: The authors consider the Cauchy-Dirichlet problem for the elliptic-parabolic equation \( b(u)_t + \mathrm{div}F(u) -\bigtriangleup u = f\) in a bounded domain. The structure condition \(b(z) = b( \widehat{z})\Rightarrow F(z) = F(\widehat{z})\) is not supposed to be satisfied. The goal of this paper is to give some partial uniqueness and continuous dependence results for the problem without the structure condition and the question of convergence of discretization methods. As in the work of \textit{K. Ammar} and \textit{P. Wittbold} [Proc. R. Soc. Edinb., Sect. A, Math. 133, No. 3, 477--496 (2003; Zbl 1077.35103)], where existence was established, monotonicity and penalization are the main tools of their study. In the case of a Lipschitz continuous flux \(F\), they 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. The convergence of the \(\varepsilon\)-discretized solutions used in the semigroup approach to the problem is proved, and convergence of a monotone time-implicit finite volume scheme is proved also. In the case of a merely continuous flux \(F\), the authors show that the problem admits a maximal and a minimal solution.
Mathematics Subject Classification ID: 35M13
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 65M08
Mathematics Subject Classification ID: 35A02
zbMATH DE Number: 6121931
zbMATH Keywords: elliptic-parabolic equation
zbMATH Keywords: structure condition
zbMATH Keywords: convergence of approximate solutions
zbMATH Keywords: finite volume method
zbMATH Keywords: penalization
zbMATH Keywords: monotone time-implicit finite volume scheme
reviewed by: Qin Meng Zhao
MaRDI profile type: Publication
full work available at URL: s00030-011-0148-8
OpenAlex ID: W2032473903
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DOI: 10.1007/S00030-011-0148-8
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description: scientific article; zbMATH DE number 6121931