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

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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


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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: mohamed.farhloul@umoncton.ca ´eetddeuVHaaliennnciiveenrnseiste´is,sUALuaCtmarb-5,F1393V,MATVIS-Valenciennes Cedex 9, France, e-mail: snicaise@univ-valenciennes.fr, http://www.univ-valenciennes.fr/macs/nicaise rbe´CtmaALAMis,sTV,FV,IS13-V-593-aUnisreve´tiaVedcnelnnieetesHaduauin lenciennes Cedex 9, France, e-mail: Luc.Paquet@univ-valenciennes.fr
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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, piecewiseRT0 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´ecoecientsλ it was shown that the estimator (of residual type, as here). There isecientandreliableindependentlyoftheLame´coecientλ experiments. These suggest that similar results should hold for the Stokes equations (because as the Lame´coecientλnesoOf).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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