A priori and a posteriori error estimations for the dual mixed finite element method of the

Mohamed Farhloul

Description

Niveau: Supérieur, Doctorat, Bac+8
A priori and a posteriori error estimations for the dual mixed finite element method of the Navier-Stokes problem M. Farhloul ?, S. Nicaise†, L. Paquet‡ May 1, 2007 Abstract This paper is concerned with a dual mixed formulation of the Navier-Stokes system in a polygonal domain of the plane with Dirichlet boundary conditions and its numerical approximation. The gradient tensor, a quantity of practical interest, is introduced as a new un- known. The problem is then approximated by a mixed finite element method. Quasi-optimal a priori error estimates are obtained. These a priori error estimates, an abstract nonlinear theory (similar to [40]) and a posteriori estimates for the Stokes system from [29] lead to an a posteriori error estimate for the Navier-Stokes system. 1 Introduction Any solution of the Navier-Stokes equations in polygonal domains has in general corner singularities [21, 34, 28]. Hence standard numerical methods lose accuracy on quasi-uniform meshes, and locally refined meshes are necessary to restore the optimal ?Universite de Moncton, Departement de Mathematiques et de Statistique, Moncton, N.B., E1A 3E9, Canada, e-mail: †Universite de Valenciennes et du Hainaut Cambresis, LAMAV, ISTV, F-59313 - Valenciennes Cedex 9, France, e-mail: snicaise@univ-valenciennes.

• univ-valenciennes


• ∂?12 ∂x2


• compressible navier-stokes


• mixed formulation


• navier stokes equations


• dirich- let boundary


• †universite de valenciennes et du hainaut cambresis

Sujets

• Paquet
• Navier?Stokes equations

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 17
• Langue: English

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priori and a posteriori error estimations for dual mixed finite element method of the Navier-Stokes problem

M. Farhloul , ∗ S. Nicaise, † L. Paquet ‡

May 1, 2007

Abstract

This paper is concerned with a dual mixed formulation of the Navier-Stokes system in a polygonal domain of the plane with Dirichlet boundary conditions and its numerical approximation. The gradient tensor, a quantity of practical interest, is introduced as a new un- known. The problem is then approximated by a mixed finite element

method. Quasi-optimal a priori error estimates are obtained. These a priori error estimates, an abstract nonlinear theory (similar to [40]) and a posteriori estimates for the Stokes system from [29] lead to an a posteriori error estimate for the Navier-Stokes system.

Introduction

the

Any solution of the Navier-Stokes equations in polygonal domains has in general corner singularities [21, 34, 28]. Hence standard numerical methods lose accuracy on quasi-uniform meshes, and locally refined meshes are necessary to restore the optimal

∗ ton,Moncuq,esiittStateedesqutima´ethMadetnemetrape´D,notinevsrtie´edoMcnU N.B., E1A 3E9, Canada, e-mail: [e-mail] † ´eetddeuVHaaliennnciiveenrnseiste´is,sUALuaCtmarb-5,F1393V,MATVIS- Valenciennes Cedex 9, France, e-mail: [e-mail], http://www.univ- valenciennes.fr/macs/nicaise ‡ rbe´CtmaALAMis,sTV,FV,IS13-V-593-aUnisreve´tiaVedcnelnnieetesHaduauin lenciennes Cedex 9, France, e-mail: [e-mail]

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order of convergence. Standard finite element methods for second order elliptic operators, the Stokes or the Navier-Stokes system with corner singularities (and mixed boundary conditions) have been analyzed in [3, 4, 6, 5, 8, 10, 12, 35, 34], where it is shown that the use of appropriate a priori refined meshes near the singular points allows to restore optimal order of convergence. Mixed methods for the Stokes and Navier-Stokes with Dirichlet boundary conditions were initiated in [24, 25]. Similarly a mixed method for the Boussinesq equations (coupling between the Navier-Stokes equations and the heat equation) with Dirichlet boundary condition on the velocity field and corner singularities is analyzed in [28] (some geometrical restrictions were made due to the mixed boundary conditions on the temperature). More recently a posteriori error analyses were developed for standard variational formulation of the above mentioned problems [1, 7, 16, 18, 38, 39, 40] where some estimators are introduced and proved to be reliable and efficient. Similar results for the dual mixed formulation of second order operators are obtained in [2, 13, 15, 33, 41], while for the Stokes system we may cite [29]. Our goal is here to consider the stationary Navier-Stokes equations with Dirich- let boundary condition in a two-dimensional polygonal domain and to approximate them by a dual mixed finite element method. Our method introduces as a new unknown (of physical interest) the gradient tensorru. The poor regularity of any solution forces us to use appropriate Banach spaces and then to introduce an ap- propriate mixed formulation of the problem. We next consider some discretization of our mixed formulation by using some mixed finite elements developed in [27]. Namely the approximation spaces are piecewise constant for the velocity, piecewise constant for the pressure, piecewiseRT 0  Wefor each line of the gradient tensor. then prove that the discrete mixed formulation has at least one solution near any nonsingular solution of the Navier-Stokes equations. Furthermore using some inter- polation error estimates we show quasi-optimal a priori error estimates. Finally our a posteriori error analysis relies on these a priori error estimates, a modified ver- sionofanabstractnonlineartheorydevelopedbyVerfu¨rth[40]andonaposteriori estimate for the Stokes system that we obtained in [29]. Let us mention that in [11] numerical experiments have been performed for a symmetricdualmixedmethodfortheLame´systeminanL-shaped domain for large Lam´ecoefficientsλ it was shown that the estimator (of residual type, as here). There isefficientandreliableindependentlyoftheLame´coefficientλ experiments. These suggest that similar results should hold for the Stokes equations (because as the Lame´coefficientλnesoOf).eSthketoheLargeteslaecomsdottmnesyet´mseb course, for the Navier-Stokes equations, we have the additional difficulty due to the nonlinear convection term, but we think that we could treat it by using a Newton- Galerkin scheme as in [26]. We plan to make such numerical experiments in the

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