Analysis of Mixed Finite Element Methods for the Stokes Problem: A Unified Approach

Stenberg, Rolf

doi:10.2307/2007557

Cited by 84 publications

(121 citation statements)

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“…The analysis relies heavily on [18], but an important new element is the localization procedure formulated in Lemmas 4.3, 5.1 and 5.2. The idea behind Lemmas 5.1 and 5.2 is in many ways similar to that underlying the macro-element technique and the corresponding local test for stability found in [3] or [14]. Much of the rest of the proof of the main theorem consists of verifying that the constants in various of the estimâtes found in [18] scale appropriately with the mesh size.…”

Section: Introductionmentioning

confidence: 93%

“…We shall make use of this fact in the case p = 3 of our proof. The proof of Theorem 5.1 relies heavily on the analysis of [18], but an added new element is the localization procedure which has certain similarities to the macro-element concept found in [3], [14] ; however, our triangulations are quite arbitrary, except for the assumption of quasiuniformity. LEMMA i.e., the inequality in (5.2) holds for/» = 3.…”

Section: The Main Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials

Scott¹,

Vogelius²

1985

ESAIM: M2AN

328

290

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Section: Introductionmentioning

confidence: 93%

Section: The Main Theoremmentioning

confidence: 99%

Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials

Scott¹,

Vogelius²

1985

ESAIM: M2AN

328

290

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“…Following the ideas in [39] and assuming a family of macro elements which is equivalent to a reference macro element, one can show that (2.1) reduces to show that…”

Section: Remark 24 Note That By Settingmentioning

confidence: 99%

A unified convergence analysis for local projection stabilisations applied to the Oseen problem

Matthies¹,

Skrzypacz²,

Tobiska³

2007

ESAIM: M2AN

170

183

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Abstract. The discretisation of the Oseen problem by finite element methods may suffer in general from two shortcomings. First, the discrete inf-sup (Babuška-Brezzi) condition can be violated. Second, spurious oscillations occur due to the dominating convection. One way to overcome both difficulties is the use of local projection techniques. Studying the local projection method in an abstract setting, we show that the fulfilment of a local inf-sup condition between approximation and projection spaces allows to construct an interpolation with additional orthogonality properties. Based on this special interpolation, optimal a-priori error estimates are shown with error constants independent of the Reynolds number. Applying the general theory, we extend the results of Braack and Burman for the standard two-level version of the local projection stabilisation to discretisations of arbitrary order on simplices, quadrilaterals, and hexahedra. Moreover, our general theory allows to derive a novel class of local projection stabilisation by enrichment of the approximation spaces. This class of stabilised schemes uses approximation and projection spaces defined on the same mesh and leads to much more compact stencils than in the two-level approach. Finally, on simplices, the spectral equivalence of the stabilising terms of the local projection method and the subgrid modelling introduced by Guermond is shown. This clarifies the relation of the local projection stabilisation to the variational multiscale approach.Mathematics Subject Classification. 65N12, 65N30, 76D05.

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“…The analysis for the stability of the macro-element type P k -P k−1 mixed element is based on a framework of Stenberg [11]. The only work left to do is to show a macro-element condition (see (4) below).…”

Section: Introductionmentioning

confidence: 99%

A new family of stable mixed finite elements for the 3D Stokes equations

2004

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Abstract. A natural mixed-element approach for the Stokes equations in the velocity-pressure formulation would approximate the velocity by continuous piecewise-polynomials and would approximate the pressure by discontinuous piecewise-polynomials of one degree lower. However, many such elements are unstable in 2D and 3D. This paper is devoted to proving that the mixed finite elements of this P k -P k−1 type when k ≥ 3 satisfy the stability conditionthe Babuška-Brezzi inequality on macro-tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedra. This type of mesh simplifies the implementation since it has no restrictions on the initial mesh. The new element also suits the multigrid method.

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Analysis of Mixed Finite Element Methods for the Stokes Problem: A Unified Approach

Cited by 84 publications

References 0 publications

Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials

Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials

A unified convergence analysis for local projection stabilisations applied to the Oseen problem

A new family of stable mixed finite elements for the 3D Stokes equations

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