On the stability of approximations for the Stokes problem using different finite element spaces for each component of the velocity

Guillén-González, Francisco; Galván, José Rafael Rodríguez

doi:10.1016/j.apnum.2015.07.002

Cited by 6 publications

(4 citation statements)

References 16 publications

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“…Numerical simulations suggest that (IS) V h also holds in unstructured meshes and therefore (P 2 , P 1 ) -P 1 would be stable. Similar results were obtained for (P 1,b , P 1 ) -P 1 (bubble enriching of U h ) and also some generalizations to 3D domains; see [14] for more details.…”

Section: Recent Results About Nonintegral Mixed Formulationssupporting

confidence: 82%

“…On the other hand, for (P 2 , P 1 ) -P 1 FE, (IS) P h holds in uniformly unstructured meshes, as is proved in [14] applying Stokes stability results about unequal approximations for U h and V h . Numerical simulations suggest that (IS) V h also holds in unstructured meshes and therefore (P 2 , P 1 ) -P 1 would be stable.…”

Section: Recent Results About Nonintegral Mixed Formulationsmentioning

confidence: 89%

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Stabilized Schemes for the Hydrostatic Stokes Equations

Guillén-González¹,

Galván²

2015

SIAM J. Numer. Anal.

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Some new stable finite element (FE) schemes are presented for the hydrostatic Stokes system or primitive equations of the ocean. It is known that the stability of the mixed formulation approximation for primitive equations requires the well-known Ladyzhenskaya-Babuška-Brezzi condition related to the Stokes problem and an extra inf-sup condition relating the pressure and the vertical velocity [F. Guillén-González and J. R. Rodríguez-Galván, Numer. Math., 130 (2015), pp. 225-256]. The main goal of this paper is to avoid this extra condition by adding a residual stabilizing term to the vertical momentum equation. Then, the stability for Stokes-stable FE combinations is extended to the primitive equations and some error estimates are provided using Taylor-Hood P 2 -P 1 or minielement (P 1 +bubble)-P 1 FE approximations, showing the optimal convergence rate in the P 2 -P 1 case. These results are also extended to the anisotropic (nonhydrostatic) problem. On the other hand, by adding another residual term to the continuity equation, a better approximation of the vertical derivative of pressure is obtained. In this case, stability and error estimates including this better approximation are deduced, where optimal convergence rate is deduced in the (P 1 +bubble)-P 1 case. Finally, some numerical experiments are presented supporting previous results.

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Section: Recent Results About Nonintegral Mixed Formulationssupporting

confidence: 82%

Section: Recent Results About Nonintegral Mixed Formulationsmentioning

confidence: 89%

Stabilized Schemes for the Hydrostatic Stokes Equations

Guillén-González¹,

Galván²

2015

SIAM J. Numer. Anal.

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“…The aim of this paper is to establish error estimates for u h and p h in the framework of analytic semigroup theory, as preliminaries for the numerical analysis of the primitive equations. Although there are several results available on the finite element method for the steady hydrostatic Stokes equations (e.g., [12,13,14,15]), there are no results for the non-stationary case, to the best of our knowledge.…”

Section: Introductionmentioning

confidence: 99%

On the finite element approximation for non-stationary saddle-point problems

Kemmochi

2018

Japan J. Indust. Appl. Math.

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In this paper, we present a numerical analysis of the hydrostatic Stokes equations, which are linearization of the primitive equations describing the geophysical flows of the ocean and the atmosphere. The hydrostatic Stokes equations can be formulated as an abstract non-stationary saddle-point problem, which also includes the non-stationary Stokes equations. We first consider the finite element approximation for the abstract equations with a pair of spaces under the discrete inf-sup condition. The aim of this paper is to establish error estimates for the approximated solutions in various norms, in the framework of analytic semigroup theory. Our main contribution is an error estimate for the pressure with a natural singularity term t −1 , which is induced by the analyticity of the semigroup. We also present applications of the error estimates for the finite element approximations of the non-stationary Stokes and the hydrostatic Stokes equations.

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“…Unfortunately, standard Stokes FE like Taylor-Hood P 2 -P 1 or (P 1 +bubble)-P 1 do not satisfy (IS) V . Thus different FE must be considered (for instance, by approximation of vertical velocity in a space other than horizontal velocity, see [GGRG15a,GGRG16]).…”

Section: Introductionmentioning

confidence: 99%

Stable Discontinuous Galerkin Approximations for the Hydrostatic Stokes Equations

Guillén-González

Redondo-Neble

Galván

2017

Computational Mathematics, Numerical Analysis and Applications

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We propose a Discontinuous Galerkin (DG) scheme for the numerical solution of the Hydrostatic Stokes equations in Oceanography. This new scheme is based on the introduction of the symmetric interior penalty (SIP) technique for the Hydrostatic Stokes mixed variational formulation. Recent research showed that stability of the mixed formulation of Primitive Equations requires LBB (Ladyzhenskaya-Babuška-Brezzi) inf-sup condition and an extra hydrostatic inf-sup restriction relating the pressure and the vertical velocity. This hydrostatic inf-sup condition invalidates usual Stokes continuous finite elements like Taylor-Hood P 2 /P 1 or bubble P 1 b /P 1 . Here we consider P k /P k discontinuous finite elements and, using adequate LBB-like and hydrostatic discrete inf-sup conditions we can demonstrate stability of the SIP DG scheme in the natural energy norm for this problem. Finally, according numerical tests are provided.

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On the stability of approximations for the Stokes problem using different finite element spaces for each component of the velocity

Cited by 6 publications

References 16 publications

Stabilized Schemes for the Hydrostatic Stokes Equations

Stabilized Schemes for the Hydrostatic Stokes Equations

On the finite element approximation for non-stationary saddle-point problems

Stable Discontinuous Galerkin Approximations for the Hydrostatic Stokes Equations

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