Convergence of approximate solutions to an elliptic-parabolic equation without the structure condition
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.
monotone time-implicit finite volume scheme
scientific article; zbMATH DE number 6121931
scientific article