A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems

Carstensen, Carsten; Funken, Stefan A.

doi:10.1090/s0025-5718-00-01264-3

Cited by 63 publications

(39 citation statements)

References 21 publications

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“…More interesting examples such as higherorder schemes, the application to the Stokes problem or the Navier-Lamé equations without incompressibility-locking will appear elsewhere [BC,CF2,CF3,CF4,CF5].…”

Section: Introductionmentioning

confidence: 99%

Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM

2002

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Abstract. Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing. In this paper, their reliablility is shown for conforming, nonconforming, and mixed low order finite element methods in a model situation: the Laplace equation with mixed boundary conditions. Emphasis is on possibly unstructured grids, nonsmoothness of exact solutions, and a wide class of averaging techniques. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides.

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Section: Introductionmentioning

confidence: 99%

Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM

2002

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“…It is by no means obvious that averaging concerns the fluxes and the gradients simultaneously. The positive examples in [CBJ,CF3,CF4,BC2,CA] may be seen as exceptions. In general, the flux and the gradient approximations may be averaged separately.…”

Section: This Is a Dirichlet Boundary Conditionmentioning

confidence: 99%

“…That is, p h is the piecewise polynomial but globally discontinuous elementwise gradient of the finite element displacement approximations u h or a discrete flux variable (for a mixed FEM) that approximates the unknown exact flux p. It is the aim of a posteriori error control to bound the error p − p h L 2 (Ω) from above and below by computable estimators [AO, BS, V]. It has recently been proven for several examples [CB,BC1,CF3,CF4] that the error p − p h L 2 (Ω) in…”

Section: Introductionmentioning

confidence: 99%

All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable

Carstensen

2003

Math. Comp.

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Abstract. All first-order averaging or gradient-recovery operators for lowestorder finite element methods are shown to allow for an efficient a posteriori error estimation in an isotropic, elliptic model problem in a bounded Lipschitz domain Ω in R d . Given a piecewise constant discrete flux p h ∈ P h (that is the gradient of a discrete displacement) as an approximation to the unknown exact flux p (that is the gradient of the exact displacement), recent results verify efficiency and reliability ofin the sense that η M is a lower and upper bound of the flux error p−p h L 2 (Ω) up to multiplicative constants and higher-order terms. The averaging space Q h consists of piecewise polynomial and globally continuous finite element functions in d components with carefully designed boundary conditions. The minimal value η M is frequently replaced by some averaging operator A : P h → Q h applied within a simple post-processing to p h . The result q h := Ap h ∈ Q h provides a reliable error bound withThis paper establishes η A ≤ C eff η M and so equivalence of η M and η A . This implies efficiency of η A for a large class of patchwise averaging techniques which includes the ZZ-gradient-recovery technique. The bound C eff ≤ 3.88 established for tetrahedral P 1 finite elements appears striking in that the shape of the elements does not enter: The equivalence η A ≈ η M is robust with respect to anisotropic meshes. The main arguments in the proof are Ascoli's lemma, a strengthened Cauchy inequality, and elementary calculations with mass matrices.

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“…In particular, for the Stokes problem, M. Ainsworth and J. Oden [2], D. Kay and D. Silvester [23], C. Cartensen and S. A. Fuken [9] and R. Verfurth [31] introduced several error estimators and provided that they are equivalent to the energy norm of the errors. Other works for the stationary Navier-Stokes problem have been introduced in [27], [32], [22], [34], [1], [26], [3].…”

Section: Introductionmentioning

confidence: 99%

An a Posteriori Error Estimate for Mixed Finite Element Approximations of the Navier-Stokes Equations

Elakkad¹,

Elkhalfi²,

Guessous³

2011

Journal of the Korean Mathematical Society

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Abstract. In this work, a numerical solution of the incompressible Navier-Stokes equations is proposed. The method suggested is based on an algorithm of discretization by mixed finite elements with a posteriori error estimation of the computed solutions. In order to evaluate the performance of the method, the numerical results are compared with some previously published works or with others coming from commercial code like Adina system.

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A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems

Cited by 63 publications

References 21 publications

Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM

Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM

All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable

An a Posteriori Error Estimate for Mixed Finite Element Approximations of the Navier-Stokes Equations

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