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

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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† 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.

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singular exponents

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sobolev spaces

†universite de valenciennes et du hainaut cambresis

standard dual

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Paquet Sobolev space

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: snicaise@univ-valenciennes.fr, http://www.univ-valenciennes.fr/macs/nicaise †Vale´edersitniveaHniteudnnsecneiLAs,si´ebramtCau31395-F,VTSI,VAMV--aU lenciennes Cedex 9, France, e-mail: Luc.Paquet@univ-valenciennes.fr, ‡nUvie´edretsaranFian,Factsoasede´tlusecneicS48.1.P,BaranFi6,Ma1,-datsan30oa gascar, e-mail: raﬁlipo@wanadoo.mg

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