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

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

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25 pages
English
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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.

  • 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


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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
Ralipojaona
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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