A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs

Boris Andreianov (1) , Clément Cancès (2, 3) , Ayman Moussa (2)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 LJLL - Laboratoire Jacques-Louis Lions
3 RAPSODI - Reliable numerical approximations of dissipative systems

We propose a discrete functional analysis result suitable for proving compactness in the framework of fully discrete approximations of strongly degenerate parabolic problems. It is based on the original exploitation of a result related to compensated compactness rather than on a classical estimate on the space and time translates in the spirit of Simon (Ann. Mat. Pura Appl. 1987). Our approach allows to handle various numerical discretizations both in the space variables and in the time variable. In particular, we can cope quite easily with variable time steps and with multistep time differentiation methods like, e.g., the backward differentiation formula of order 2 (BDF2) scheme. We illustrate our approach by proving the convergence of a two-point flux Finite Volume in space and BDF2 in time approximation of the porous medium equation.

Mots clés: degenerate parabolic equations; strong compactness; gradient scheme; weakly convergent sequences

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

Boris Andreianov, Clément Cancès, Ayman Moussa. A nonlinear time compactness result and applications to discretization of degenerate parabolic-elliptic PDEs. Journal of Functional Analysis, 2017, 273 (12), pp.3633-3670. ⟨10.1016/j.jfa.2017.08.010⟩. ⟨hal-01142499⟩