Finite Element Approximation of the Navier-Stokes Equations

Girault, Vivette; Raviart, Pierre Arnaud

doi:10.1007/bfb0063447

Cited by 933 publications

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“…, p and vanish on Γ ∞ . It is an easy consequence of Lemma 5.3 (Chapter 3) of [5] that if q ∈ Q h satisfies ∇×q = 0, then q = ∇p for some p ∈S h .…”

Section: Lemma 31 Letp H Be Defined Bymentioning

confidence: 99%

Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem

Bramble

Pasciak

2008

Math. Comp.

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Abstract. In our paper [Math. Comp. 76, 2007, 597-614] we considered the acoustic and electromagnetic scattering problems in three spatial dimensions. In particular, we studied a perfectly matched layer (PML) approximation to an electromagnetic scattering problem. We demonstrated both the solvability of the continuous PML approximations and the exponential convergence of the resulting solution to the solution of the original acoustic or electromagnetic problem as the layer increased.In this paper, we consider finite element approximation of the truncated PML electromagnetic scattering problem. Specifically, we consider approximations which result from the use of Nédélec (edge) finite elements. We show that the resulting finite element problem is stable and gives rise to quasi-optimal convergence when the mesh size is sufficiently small.

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“…, p and vanish on Γ ∞ . It is an easy consequence of Lemma 5.3 (Chapter 3) of [5] that if q ∈ Q h satisfies ∇×q = 0, then q = ∇p for some p ∈S h .…”

Section: Lemma 31 Letp H Be Defined Bymentioning

confidence: 99%

Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem

Bramble

Pasciak

2008

Math. Comp.

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“…This completes the proof of the above claim. If we further assume u ∈ W 1,3 (Ω) ∩ L ∞ (Ω), then, the regularity assumption (2.3) comes from a standard regularity estimate [13] for the Stokes equations. Now we are ready to present our second and main result:…”

Section: Resultsmentioning

confidence: 99%

“…See the classic mixed methods [13,2,12], the stabilized methods proposed in [17,16,19] and the DG methods [1,18,7,14,28,27,5,8,9,10,21]. An IP-like method and a compact discontinuous Galerkin (CDG) method were introduced in [22].…”

Section: Introductionmentioning

confidence: 99%

A superconvergent HDG method for the incompressible Navier–Stokes equations on general polyhedral meshes

Qiu

Shi

2016

IMA J Numer Anal

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Abstract. We present a superconvergent hybridizable discontinuous Galerkin (HDG) method for the steady-state incompressible Navier-Stokes equations on general polyhedral meshes. For arbitrary conforming polyhedral mesh, we use polynomials of degree k + 1, k, k to approximate the velocity, velocity gradient and pressure, respectively. In contrast, we only use polynomials of degree k to approximate the numerical trace of the velocity on the interfaces. Since the numerical trace of the velocity field is the only globally coupled unknown, this scheme allows a very efficient implementation of the method. For the stationary case, and under the usual smallness condition for the source term, we prove that the method is well defined and that the global L 2 -norm of the error in each of the above-mentioned variables and the discrete H 1 -norm of the error in the velocity converge with the order of k + 1 for k ≥ 0. We also show that for k ≥ 1, the global L 2 -norm of the error in velocity converges with the order of k + 2. From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this method achieves optimal convergence for all the above-mentioned variables in L 2 -norm for k ≥ 0, superconvergence for the velocity in the discrete H 1 -norm without postprocessing for k ≥ 0, and superconvergence for the velocity in L 2 -norm without postprocessing for k ≥ 1.

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“…We could prove independently of Section 2 similar results to Theorem 2.5 and Theorem 2.6 by adapting the div-curl result of Theorem 2.11 to the Stokes problem. For the sake of simplicity we have preferred applying the results of Section 2 by using the classical representation of a divergence free function as the rotated gradient of a stream function (see, e.g., [19]). This representation allows us to transform a Stokes equation into a simply connected domain by a fourth-order plate equation.…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…Observe that if ω is a Lipschitz simply connected open subset of Ω and u n ∈ H 1 div (ω) (see, e.g., [19]) there exists w n ∈ H 2 (ω) with zero mean value in ω such that u n = J∇w n . Moreover, if u n satisfies (3.4), (3.5), then the sequence w n satisfies Du n = JD 2 w n and B n Du n = JA n D 2 w n .…”

Section: Statement Of the Resultsmentioning

confidence: 99%

Homogenization of Stiff Plates and Two-Dimensional High-Viscosity Stokes Equations

Briane

Casado–Díaz

2012

Arch Rational Mech Anal

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The paper deals with the homogenization of rigid heterogeneous plates. Assuming that the coefficients are equi-bounded in L 1 , we prove that the limit of a sequence of plate equations remains a plate equation which involves a strongly local linear operator acting on the second gradients. This compactness result is based on a div-curl lemma for fourthorder equations. On the other hand, using an intermediate stream function we deduce from the plates case a similar result for high-viscosity Stokes equations in dimension two, so that the nature of the Stokes equation is preserved in the homogenization process. Finally, we show that the L 1 -boundedness assumption cannot be relaxed. Indeed, in the case of the Stokes equation the concentration of one very rigid strip on a line induces the appearance of second gradient terms in the limit problem, which violates the compactness result obtained under the L 1 -boundedness condition.

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Finite Element Approximation of the Navier-Stokes Equations

Cited by 933 publications

References 0 publications

Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem

Analysis of a finite element PML approximation for the three dimensional time-harmonic Maxwell problem

A superconvergent HDG method for the incompressible Navier–Stokes equations on general polyhedral meshes

Homogenization of Stiff Plates and Two-Dimensional High-Viscosity Stokes Equations

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