First vorticity–velocity–pressure numerical scheme for the Stokes problem

Dubois, François; Salaün, Michel; Salmon, Stéphanie

doi:10.1016/s0045-7825(03)00377-3

Cited by 26 publications

(36 citation statements)

References 16 publications

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“…We have shown theoretically and numerically that by this way we obtain convergence with optimal rate in some cases on the quadratic norm of the vorticity. We use the same method for a vorticity-velocity-pressure formulation of the Stokes problem that allows more general boundary conditions [3,5]. The results are not published yet but they are really satisfying as, once again, the solution and the convergence of the method are improved.…”

Section: Discussionmentioning

confidence: 99%

“…The convergence for the quadratic norm of the vorticity is of order ͌ h , where h is the maximum diameter of the triangles in the mesh, and of order h 1Ϫ for the H 1 -norm of the stream function ( is an arbitrary strictly positive real). Notice that in the case of structured meshes, the usual scheme gives optimal numerical results (see e.g., [3,5]) and moreover, superconvergence can be observed [2].…”

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confidence: 99%

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Coupling harmonic functions‐finite elements for solving the stream function‐vorticity Stokes problem

Abboud¹,

Salaün²,

Salmon³

2004

Numerical Methods Partial

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We consider the bidimensional Stokes problem for incompressible fluids in stream function‐vorticity form. The classical finite element method of degree one usually used does not allow the vorticity on the boundary of the domain to be computed satisfactorily when the meshes are unstructured and does not converge optimally. To better approach the vorticity along the boundary, we propose that harmonic functions obtained by integral representation be used. Numerical results are very satisfactory, and we prove that this new numerical scheme leads to an optimal convergence rate of order 1 for the natural norm of the vorticity and, under higher regularity assumptions, from 3/2 to 2 for the quadratic norm of the vorticity. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.

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Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

Coupling harmonic functions‐finite elements for solving the stream function‐vorticity Stokes problem

Abboud¹,

Salaün²,

Salmon³

2004

Numerical Methods Partial

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“…In view of the finite element discretization and according to the ideas in [2,14,15,27], we now propose a modified formulation of problem (2.5), which relies on the introduction of a new unknown: the vorticity ω associated with u in Ω F .…”

Section: Another Formulationmentioning

confidence: 99%

Coupling Darcy and Stokes equations for porous media with cracks

Bernardi¹,

Hecht²,

Pironneau³

2005

ESAIM: M2AN

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Abstract. In order to handle the flow of a viscous incompressible fluid in a porous medium with cracks, the thickness of which cannot be neglected, we consider a model which couples the Darcy equations in the medium with the Stokes equations in the cracks by a new boundary condition at the interface, namely the continuity of the pressure. We prove that this model admits a unique solution and propose a mixed formulation of it. Relying on this formulation, we describe a finite element discretization and derive a priori and a posteriori error estimates. We present some numerical experiments that are in good agreement with the analysis.Mathematics Subject Classification. 65N30, 65N50, 76D07, 76S05.

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“…A somewhat different approach has been presented in [22], where the problem is written as a system of first order equations and the resulting variables are discretized in terms of P 1 À RT 0 À P 0 elements. In that contribution the authors report optimal convergence for the three fields when structured meshes were employed, whereas on unstructured meshes and in the case of general boundary conditions, the obtained results were inaccurate.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the observed convergence was not optimal, while vorticity and pressure fields were not well approximated, specially on the boundaries. More recently, a stabilization procedure was introduced in [40], mainly to improve the convergence behavior of the method presented in [22]. This strategy is based on adding bubble functions along a part of the boundary.…”

Section: Introductionmentioning

confidence: 99%

An augmented mixed finite element method for the vorticity–velocity–pressure formulation of the Stokes equations

Anaya

Mora

Ruiz–Baier

2013

Computer Methods in Applied Mechanics and Engineering

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a b s t r a c tThis paper deals with the numerical approximation of the stationary two-dimensional Stokes equations, formulated in terms of vorticity, velocity and pressure, with non-standard boundary conditions. Here, by introducing a Galerkin least-squares term, we end up with a stabilized variational formulation that can be recast as a twofold saddle point problem. We propose two families of mixed finite elements to solve the discrete problem, in the first family, the unknowns are approximated by piecewise continuous and quadratic elements, Brezzi-Douglas-Marini, and piecewise constant finite elements, respectively, while in the second family, the unknowns are approximated by piecewise linear and continuous, Raviart-Thomas, and piecewise constant finite elements, respectively. The wellposedness of the resulting continuous and discrete variational problems are studied employing an extension of the Babuška-Brezzi theory. We establish a priori error estimates in the natural norms, and we finally report some numerical experiments illustrating the behavior of the proposed schemes and confirming our theoretical findings on structured and unstructured meshes. Additional examples of cases not covered by our theory are also presented.

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First vorticity–velocity–pressure numerical scheme for the Stokes problem

Cited by 26 publications

References 16 publications

Coupling harmonic functions‐finite elements for solving the stream function‐vorticity Stokes problem

Coupling harmonic functions‐finite elements for solving the stream function‐vorticity Stokes problem

Coupling Darcy and Stokes equations for porous media with cracks

An augmented mixed finite element method for the vorticity–velocity–pressure formulation of the Stokes equations

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