Improved Crouzeix-Raviart scheme for the Stokes and Navier-Stokes problem
The resolution of the incompressible Navier-Stokes equations is tricky, and it is well known that one of the major issue is to approach the space: H1(Ω)∩ H(div 0; Ω) := {v ∈ H1(Ω) : div v = 0}. H 1 (Ω)∩ H ( div 0 ;Ω):= { v ∈ H 1 (Ω): div v = 0 }. $$ \mathbf{H}^1(\Omega) \cap \...
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| Language: | English |
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EDP Sciences
2024-01-01
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| Series: | ESAIM: Proceedings and Surveys |
| Online Access: | https://www.esaim-proc.org/articles/proc/pdf/2024/01/proc2407602.pdf |
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| author | Chénier Eric Jamelot Erell Le Potier Christophe Peitavy Andrew |
| author_facet | Chénier Eric Jamelot Erell Le Potier Christophe Peitavy Andrew |
| author_sort | Chénier Eric |
| collection | DOAJ |
| description | The resolution of the incompressible Navier-Stokes equations is tricky, and it is well known that one of the major issue is to approach the space:
H1(Ω)∩ H(div 0; Ω) := {v ∈ H1(Ω) : div v = 0}.
H
1
(Ω)∩
H
(
div
0
;Ω):=
{
v
∈
H
1
(Ω):
div
v
=
0
}.
$$ \mathbf{H}^1(\Omega) \cap \mathbf{H}(\operatorname{div} 0 ; \Omega):=\left\{\mathbf{v} \in \mathbf{H}^1(\Omega): \operatorname{div} \mathbf{v}=0\right\}. $$
The non-conforming Crouzeix-Raviart finite element are convenient since they induce local mass conservation. Moreover they are such that the stability constant of the Fortin operator is equal to 1. This implies that they can easily handle anisotropic mesh. However spurious velocities may appear and damage the approximation.
We propose a scheme here that allows one to reduce the spurious velocities. It is based on a new discretisation for the gradient of pressure based on the symmetric MPFA scheme (finite volume MultiPoint Flux Approximation). |
| format | Article |
| id | doaj-art-4ec07f3c2f9e4c7094e6cb6d7a4b41e0 |
| institution | Kabale University |
| issn | 2267-3059 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | EDP Sciences |
| record_format | Article |
| series | ESAIM: Proceedings and Surveys |
| spelling | doaj-art-4ec07f3c2f9e4c7094e6cb6d7a4b41e02024-12-06T10:47:25ZengEDP SciencesESAIM: Proceedings and Surveys2267-30592024-01-0176203410.1051/proc/202476020proc2407602Improved Crouzeix-Raviart scheme for the Stokes and Navier-Stokes problemChénier Eric0Jamelot Erell1Le Potier Christophe2Peitavy Andrew3Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, UMR 8208, MSMEUniversité Paris-Saclay, CEA, Service de Thermo-hydraulique et de Mécanique des FluidesUniversité Paris-Saclay, CEA, Service de Thermo-hydraulique et de Mécanique des FluidesUniversité Paris-Saclay, CEA, Service de Thermo-hydraulique et de Mécanique des FluidesThe resolution of the incompressible Navier-Stokes equations is tricky, and it is well known that one of the major issue is to approach the space: H1(Ω)∩ H(div 0; Ω) := {v ∈ H1(Ω) : div v = 0}. H 1 (Ω)∩ H ( div 0 ;Ω):= { v ∈ H 1 (Ω): div v = 0 }. $$ \mathbf{H}^1(\Omega) \cap \mathbf{H}(\operatorname{div} 0 ; \Omega):=\left\{\mathbf{v} \in \mathbf{H}^1(\Omega): \operatorname{div} \mathbf{v}=0\right\}. $$ The non-conforming Crouzeix-Raviart finite element are convenient since they induce local mass conservation. Moreover they are such that the stability constant of the Fortin operator is equal to 1. This implies that they can easily handle anisotropic mesh. However spurious velocities may appear and damage the approximation. We propose a scheme here that allows one to reduce the spurious velocities. It is based on a new discretisation for the gradient of pressure based on the symmetric MPFA scheme (finite volume MultiPoint Flux Approximation).https://www.esaim-proc.org/articles/proc/pdf/2024/01/proc2407602.pdf |
| spellingShingle | Chénier Eric Jamelot Erell Le Potier Christophe Peitavy Andrew Improved Crouzeix-Raviart scheme for the Stokes and Navier-Stokes problem ESAIM: Proceedings and Surveys |
| title | Improved Crouzeix-Raviart scheme for the Stokes and Navier-Stokes problem |
| title_full | Improved Crouzeix-Raviart scheme for the Stokes and Navier-Stokes problem |
| title_fullStr | Improved Crouzeix-Raviart scheme for the Stokes and Navier-Stokes problem |
| title_full_unstemmed | Improved Crouzeix-Raviart scheme for the Stokes and Navier-Stokes problem |
| title_short | Improved Crouzeix-Raviart scheme for the Stokes and Navier-Stokes problem |
| title_sort | improved crouzeix raviart scheme for the stokes and navier stokes problem |
| url | https://www.esaim-proc.org/articles/proc/pdf/2024/01/proc2407602.pdf |
| work_keys_str_mv | AT cheniereric improvedcrouzeixraviartschemeforthestokesandnavierstokesproblem AT jameloterell improvedcrouzeixraviartschemeforthestokesandnavierstokesproblem AT lepotierchristophe improvedcrouzeixraviartschemeforthestokesandnavierstokesproblem AT peitavyandrew improvedcrouzeixraviartschemeforthestokesandnavierstokesproblem |