Convergence and a posteriori error analysis for energy-stable finite element approximations of degenerate parabolic equations

Cancès, Clément; Nabet, Flore; Vohralík, Martin

Document type :

Compte-rendu et recension critique d'ouvrage

DOI :

Title :

Convergence and a posteriori error analysis for energy-stable finite element approximations of degenerate parabolic equations

Author(s) :

Cancès, Clément [Auteur]
Reliable numerical approximations of dissipative systems [RAPSODI]
Nabet, Flore [Auteur correspondant]
Centre de Mathématiques Appliquées de l'Ecole polytechnique [CMAP]
Vohralík, Martin [Auteur]
Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique [CERMICS]
Simulation for the Environment: Reliable and Efficient Numerical Algorithms [SERENA]

Journal title :

Mathematics of Computation

Pages :

517-563

Publisher :

American Mathematical Society

Publication date :

2021

ISSN :

0025-5718

English keyword(s) :

Convergence
Degenerate parabolic equation
Equilibrated flux
Energy-stable discretization
Local conservation
Fokker-Planck equation
A posteriori error estimate

HAL domain(s) :

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

English abstract : [en]

We propose a finite element scheme for numerical approximation of degenerate parabolic problems in the form of a nonlinear anisotropic Fokker-Planck equation. The scheme is energy-stable, only involves physically motivated ...
Show more >We propose a finite element scheme for numerical approximation of degenerate parabolic problems in the form of a nonlinear anisotropic Fokker-Planck equation. The scheme is energy-stable, only involves physically motivated quantities in its definition, and is able to handle general unstructured grids. Its convergence is rigorously proven thanks to compactness arguments, under very general assumptions. Although the scheme is based on Lagrange finite elements of degree 1, it is locally conservative after a local postprocess giving rise to an equilibrated flux. This also allows to derive a guaranteed a posteriori error estimate for the approximate solution. Numerical experiments are presented in order to give evidence of a very good behavior of the proposed scheme in various situations involving strong anisotropy and drift terms.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Approche géométrique pour les écoulements en milieux poreux: théorie et numérique
Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :