Finite element approximation of the FENE-P model - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue IMA Journal of Numerical Analysis Année : 2017

## Finite element approximation of the FENE-P model

(1) , (2, 3)
1
2
3
John W Barrett
• Fonction : Auteur
• PersonId : 1005416
Sébastien Boyaval

#### Résumé

We extend our analysis on the Oldroyd-B model in Barrett and Boyaval [1] to consider the finite element approximation of the FENE-P system of equations, which models a dilute polymeric fluid, in a bounded domain $D ⊂ R d , d = 2 or 3$, subject to no flow boundary conditions. Our schemes are based on approximating the pressure and the symmetric conforma-tion tensor by either (a) piecewise constants or (b) continuous piecewise linears. In case (a) the velocity field is approximated by continuous piecewise quadratics ($d = 2$) or a reduced version, where the tangential component on each simplicial edge ($d = 2$) or face ($d = 3$) is linear. In case (b) the velocity field is approximated by continuous piecewise quadratics or the mini-element. We show that both of these types of schemes, based on the backward Euler type time discretiza-tion, satisfy a free energy bound, which involves the logarithm of both the conformation tensor and a linear function of its trace, without any constraint on the time step. Furthermore, for our approximation (b) in the presence of an additional dissipative term in the stress equation, the so-called FENE-P model with stress diffusion, we show (subsequence) convergence in the case $d = 2$, as the spatial and temporal discretization parameters tend to zero, towards global-in-time weak solutions of this FENE-P system. Hence, we prove existence of global-in-time weak solutions to the FENE-P model with stress diffusion in two spatial dimensions.

#### Domaines

Mathématiques [math] Analyse numérique [math.NA]

### Dates et versions

hal-01501197 , version 1 (03-04-2017)

### Identifiants

• HAL Id : hal-01501197 , version 1
• ARXIV :
• DOI :

### Citer

John W Barrett, Sébastien Boyaval. Finite element approximation of the FENE-P model. IMA Journal of Numerical Analysis, 2017, ⟨10.1093/imanum/drx061⟩. ⟨hal-01501197⟩

### Exporter

BibTeX TEI Dublin Core DC Terms EndNote Datacite

### Collections

366 Consultations
383 Téléchargements