Titre :

Finite elements for Wasserstein $W_p$ gradient flows

Auteur(s) :

Cancès, Clément [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Reliable numerical approximations of dissipative systems [RAPSODI]
Matthes, Daniel [Auteur]
Zentrum Mathematik [Munchen] [TUM]
Nabet, Flore [Auteur]
Centre de Mathématiques Appliquées de l'Ecole polytechnique [CMAP]
Rott, Eva-Maria [Auteur]
Zentrum Mathematik [Munchen] [TUM]

Mot(s)-clé(s) en anglais :

Wasserstein gradient flow
finite elements
nonlinear stability
convergence analysis

Discipline(s) HAL :

Mathématiques [math]/Analyse numérique [math.NA]
Mathématiques [math]/Equations aux dérivées partielles [math.AP]

Résumé en anglais : [en]

Wasserstein $\bbW_p$ gradient flows for nonlinear integral functionals of the density yield degenerate parabolic equations involving diffusion operators of $q$-Laplacian type, with $q$ being $p$'s conjugate exponent. We ...
Lire la suite >Wasserstein $\bbW_p$ gradient flows for nonlinear integral functionals of the density yield degenerate parabolic equations involving diffusion operators of $q$-Laplacian type, with $q$ being $p$'s conjugate exponent. We propose a finite element scheme building on conformal $\mathbb{P}_1$ Lagrange elements with mass lumping and a backward Euler time discretization strategy. Our scheme preserves mass and positivity while energy decays in time.Building on the theory of gradient flows in metric spaces, we further prove convergence towards a weak solution of the PDE that satisfies the energy dissipation equality. The analytical results are illustrated by numerical simulations.Lire moins >

Langue :

Anglais

Projet ANR :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL