A refined mixed finite element method for the stationary Navier Stokes equations with mixed boundary conditions using Lagrange multipliers

Luc Paquet

Description

A refined mixed finite element method for the stationary Navier-Stokes equations with mixed boundary conditions using Lagrange multipliers S. Nicaise? L. Paquet† Rafilipojaona ‡ October 2, 2006 Abstract This paper is concerned with a dual mixed formulation of the Navier-Stokes system in a polygonal domain of the plane with mixed boundary conditions and its numerical approximation. The Neumann boundary condition is imposed using a Lagrange multiplier corre- sponding to the velocity field. Moreover the strain tensor and the anti- symmetric gradient tensor (vorticity), quantities of practical interest, are introduced as new unknowns. The problem is then approximated by a mixed finite element method. Quasi-optimal error estimates are finally obtained using refined meshes near singular corners. 1 Introduction Any solution of the Navier-Stokes equations in polygonal domains has in general corner singularities [11, 30, 18]. Hence standard numerical methods lose accuracy on quasi- uniform meshes, and locally refined meshes are necessary to restore the optimal order ?Universite de Valenciennes et du Hainaut Cambresis, LAMAV, ISTV, F-59313 - Valenciennes Cedex 9, France, e-mail: , valenciennes.fr/macs/nicaise †Universite de Valenciennes et du Hainaut Cambresis, LAMAV, ISTV, F-59313 - Va- lenciennes Cedex 9, France, e-mail: Luc.Paquet@univ-valenciennes.

• univ-valenciennes


• r?d?v ?


• hence standard


• stokes problem near


• singular exponents


• neumann boundary


• sobolev spaces


• †universite de valenciennes et du hainaut cambresis


• standard dual

Sujets

• Paquet
• Sobolev space

Informations

• Publié par: pefav
• Nombre de lectures: 21
• Langue: English

Extrait

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A refined mixed finite element method for the stationary Navier-Stokes equations with mixed boundary conditions using Lagrange multipliers

S. Nicaise ∗

L. Paquet †

October 2, 2006

Rafilipojaona ‡

Abstract This paper is concerned with a dual mixed formulation of the Navier-Stokes system in a polygonal domain of the plane with mixed boundary conditions and its numerical approximation. The Neumann boundary condition is imposed using a Lagrange multiplier corre- sponding to the velocity field. Moreover the strain tensor and the anti- symmetric gradient tensor (vorticity), quantities of practical interest, are introduced as new unknowns. The problem is then approximated by a mixed finite element method. Quasi-optimal error estimates are finally obtained using refined meshes near singular corners.

Introduction

Any solution of the Navier-Stokes equations in polygonal domains has in general corner singularities [11, 30, 18]. Hence standard numerical methods lose accuracy on quasi- uniform meshes, and locally refined meshes are necessary to restore the optimal order

∗ sis,br´etCaminauudaHseteeinnelcnVade´eitrsveniU-13F,VT395-AMALSI,V Valenciennes Cedex 9, France, e-mail: [e-mail], http://www.univ- valenciennes.fr/macs/nicaise † Vale´edersitniveaHniteudnnsecneiLAs,si´ebramtCau31395-F,VTSI,VAMV--aU lenciennes Cedex 9, France, e-mail: [e-mail], ‡ nUvie´edretsaranFian,Factsoasede´tlusecneicS48.1.P,BaranFi6,Ma1,-datsan30oa gascar, e-mail: rafi[e-mail]

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