Time compactness tools for discretized evolution equations and applications to degenerate parabolic PDEs

Boris Andreianov (1)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

We discuss several techniques for proving compactness of sequences of approximate solutions to discretized evolution PDEs. While the well-known Aubin-Simon kind functional-analytic techniques were recently generalized to the discrete setting by Gallouët and Latché [15], here we discuss direct techniques for estimating the time translates of approximate solutions in the space $L^1$. One important result is the Kruzhkov time compactness lemma. Further, we describe a specific technique that relies upon the order-preservation property. Motivation comes from studying convergence of finite volume discretizations for various classes of nonlinear degenerate parabolic equations. These and other applications are briefly described.

Mots clés: Compactness; Time translates; Kruzhkov lemma; Order-preservation; Finite volumes

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Boris Andreianov. Time compactness tools for discretized evolution equations and applications to degenerate parabolic PDEs. Finite Volumes for Complex Applications VI, Jun 2011, Prague, Czech Republic. pp. 21-29, ⟨10.1007/978-3-642-20671-9_3⟩. ⟨hal-00561344v2⟩