Stabilized finite element method  for Navier--Stokes equations  with physical boundary conditions

Amara, Mohamed; Capatina-Papaghiuc, Daniela; Trujillo, David

doi:10.1090/s0025-5718-07-01929-1

Cited by 32 publications

(22 citation statements)

References 17 publications

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“…Even if the regularity results for the Stokes problem with mixed boundary conditions in a three-dimensional domain are presently unknown, this property seems likely for a convex polyhedron (see [18] for the first results in dimension 2) and can easily be extended to the nonlinear case. On the other hand, part (2) means that the solution u is locally unique, which is much weaker than the global uniqueness, see Proposition 2.8.…”

Section: Lemma 43 If Assumption 41 Holds For Anymentioning

confidence: 99%

A finite element discretization of the three-dimensional Navier–Stokes equations with mixed boundary conditions

2009

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Abstract.We consider a variational formulation of the three-dimensional Navier-Stokes equations with mixed boundary conditions and prove that the variational problem admits a solution provided that the domain satisfies a suitable regularity assumption. Next, we propose a finite element discretization relying on the Galerkin method and establish a priori and a posteriori error estimates.Mathematics Subject Classification. 65N30, 65N15, 65J15.

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Section: Lemma 43 If Assumption 41 Holds For Anymentioning

confidence: 99%

A finite element discretization of the three-dimensional Navier–Stokes equations with mixed boundary conditions

2009

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“…So, the quantity (ω, u), (ω, u) is nonnegative on the sphere S µ with radius 2 , it is a separable Hilbert space. So there exists an increasing sequence (W n ) n of finite-dimensional subspaces W n of W such that ∪ n W n is dense in W. The mapping is continuous from W n onto its dual space and satisfies…”

Section: Proposition 25mentioning

confidence: 99%

“…The numerical analysis of discretizations relying on the vorticity, velocity and pressure formulation has first been performed for finite element methods, see [21] and [2]. In the much simpler case of the Stokes problem, it has been recently extended to the case of spectral methods in [7], where spectral analogues of Nédélec's finite elements [20] are used.…”

Section: Introductionmentioning

confidence: 99%

Spectral discretization of the vorticity, velocity and pressure formulation of the Navier–Stokes equations

Azaïez¹,

Bernardi

Chorfi

2006

Numer. Math.

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We consider the Navier-Stokes equations in a two-or three-dimensional domain provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We write a variational formulation of this problem with three independent unknowns, the vorticity, the velocity and the pressure, and prove the existence of a solution for this problem. Next we propose a discretization by spectral methods which relies on this formulation. In the two-dimensional case, we prove quasi-optimal error estimates for the three unknowns. We conclude with some numerical experiments. Résumé: Nous considérons les équations de Navier-Stokes dans un domaine biou tri-dimensionnel, munies de conditions aux limites non usuelles portant sur la composante normale de la vitesse et la ou les composantes tangentielles du tourbillon. Nous écrivons une formulation variationnelle de ce problème qui comporte trois inconnues indépendantes: le tourbillon, la vitesse et la pression. Nous prouvons que ce problème admet au moins une solution. Nous proposons une discrétisation par méthodes spectrales construite à partir de cette formulation. Dans le cas bidimensionnel, nous établissons des majorations quasi-optimales de l'erreur pour les trois inconnues. Nous concluons par quelques expériences numériques.

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“…Several numerical methods exploit these properties, as for instance, different formulations based on least-squares, stabilization techniques, mixed finite elements, spectral discretizations, and hybridizable discontinuous Galerkin methods (see for instance [3,4,8,12,14,18,19,[21][22][23]26,[34][35][36], and the references therein). For the generalized Stokes problem written in velocity-vorticity-pressure variables, we mention [6] where an augmented mixed formulation based on RT k − P k+1 − P k+1 (with continuous pressure approximation) finite elements has been developed and analyzed.…”

Section: Introductionmentioning

confidence: 99%

A priori and a posteriori error analysis of a mixed scheme for the Brinkman problem

et al. 2015

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This paper deals with the analysis of new mixed finite element methods for the Brinkman equations formulated in terms of velocity, vorticity and pressure. Employing the Babuška-Brezzi theory, it is proved that the resulting continuous and discrete variational formulations are well-posed. In particular, we show that RaviartThomas elements of order k ≥ 0 for the approximation of the velocity field, piecewise continuous polynomials of degree k + 1 for the vorticity, and piecewise polynomials of degree k for the pressure, yield unique solvability of the discrete problem. On the other hand, we also show that families of finite elements based on Brezzi-DouglasMarini elements of order k +1 for the approximation of velocity, piecewise continuous polynomials of degree k +2 for the vorticity, and piecewise polynomials of degree k for the pressure ensure the well-posedness of the associated Galerkin scheme. We note that these families provide exactly divergence-free approximations of the velocity field. We establish a priori error estimates in the natural norms with constants independent B Ricardo Ruiz-Baier

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Stabilized finite element method for Navier--Stokes equations with physical boundary conditions

Cited by 32 publications

References 17 publications

A finite element discretization of the three-dimensional Navier–Stokes equations with mixed boundary conditions

A finite element discretization of the three-dimensional Navier–Stokes equations with mixed boundary conditions

Spectral discretization of the vorticity, velocity and pressure formulation of the Navier–Stokes equations

A priori and a posteriori error analysis of a mixed scheme for the Brinkman problem

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