On the stability of the $L^2$ projection in $H^1(\Omega)$

Bramble, James H.; Pasciak, Joseph E.; Steinbach, Olaf

doi:10.1090/s0025-5718-01-01314-x

Cited by 161 publications

(143 citation statements)

References 17 publications

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“…The comparatively large values of c 6 in experiments 2 and 4 seem to be caused by the sharp change of the element sizes. This observation is supported by experiment 5 which features a smoother change of the element sizes and reveals smaller values of c 6 . In view of experiment 4, the mesh assumption (A3) and (A4) seem to be of less influence on c 5 and c 6 .…”

Section: Results For Theorem 313supporting

confidence: 66%

“…This observation is supported by experiment 5 which features a smoother change of the element sizes and reveals smaller values of c 6 . In view of experiment 4, the mesh assumption (A3) and (A4) seem to be of less influence on c 5 and c 6 . The global equivalence (3.30) between the residual estimator and the ZZ estimator reads η R ∼ η Z2 .…”

Section: Results For Theorem 313supporting

confidence: 66%

“…Recent research [6,7,23] suggests that the restrictions of (A5) can be weakened; some results of the aforementioned work already apply to our setting here.…”

Section: Mesh Requirementsmentioning

confidence: 67%

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Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes

Kunert

Nicaise

2003

ESAIM: M2AN

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Abstract. We consider a posteriori error estimators that can be applied to anisotropic tetrahedral finite element meshes, i.e. meshes where the aspect ratio of the elements can be arbitrarily large. Two kinds of Zienkiewicz-Zhu (ZZ) type error estimators are derived which originate from different backgrounds. In the course of the analysis, the first estimator turns out to be a special case of the second one, and both estimators can be expressed using some recovered gradient. The advantage of keeping two different analyses of the estimators is that they allow different and partially novel investigations and results. Both rigorous analytical approaches yield the equivalence of each ZZ error estimator to a known residual error estimator. Thus reliability and efficiency of the ZZ error estimation is obtained. The anisotropic discretizations require analytical tools beyond the standard isotropic methods. Particular attention is paid to the requirements on the anisotropic mesh. The analysis is complemented and confirmed by extensive numerical examples. They show that good results can be obtained for a large class of problems, demonstrated exemplary for the Poisson problem and a singularly perturbed reaction diffusion problem.Mathematics Subject Classification. 65N15, 65N30, 65N50.

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Section: Results For Theorem 313supporting

confidence: 66%

Section: Results For Theorem 313supporting

confidence: 66%

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Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes

Kunert

Nicaise

2003

ESAIM: M2AN

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“…The quasi-uniform assumption on the mesh is used solely to guarantee that the L 2 (Ω r ) projection onto the finite element subspaceÛ h is stable when restricted H 1 (Ω r ). Stability of the projection can established under much weaker restrictions on the mesh geometry [4,6]. In this situation the proof of (3) would proceed directly as in the proof of (1).…”

Section: Lemma 34 Let {T H } H>0 Be a Quasi-regular Family Of Trianmentioning

confidence: 99%

Lagrangian and moving mesh methods for the convection diffusion equation

Chrysafinos¹,

Walkington²

2008

ESAIM: M2AN

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“…The H 1 -stability of the L 2 -projection can, however, actually be proven under much weaker conditions on the grids. Partly rather technical sufficient conditions for the H 1 -stability of the L 2 -projection are given in [7][8][9][10]. In [11], the case of piecewise linear elements in two space dimensions is treated.…”

Section: The Rayleigh-ritz Method: Framework and Assumptionsmentioning

confidence: 99%

A Short Theory of the Rayleigh–Ritz Method

Yserentant

2013

Computational Methods in Applied Mathematics

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-We present some new error estimates for the eigenvalues and eigenfunctions obtained by the Rayleigh-Ritz method, the common variational method to solve eigenproblems. The errors are bounded in terms of the error of the best approximation of the eigenfunction under consideration by functions in the ansatz space. In contrast to the classical theory, the approximation error of eigenfunctions other than the given one does not enter into these estimates. The estimates are based on a bound for the norm of a certain projection operator, e.g., in finite element methods for second order eigenvalue problems, the H 1 -norm of the L 2 -projection onto the finite element space.2010 Mathematical subject classification: 65N25, 65N15, 65N30.

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On the stability of the $L^2$ projection in $H^1(\Omega)$

Cited by 161 publications

References 17 publications

Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes

Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes

Lagrangian and moving mesh methods for the convection diffusion equation

A Short Theory of the Rayleigh–Ritz Method

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