A posteriori error estimators for nonconforming finite element methods

Dari, Enzo A.; Durán, Ricardo G.; Padra, Claudio; Vampa, Victoria

doi:10.1051/m2an/1996300403851

Cited by 127 publications

(110 citation statements)

References 12 publications

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“…We establish the lower error bound of the estimator η p K in a more or less standard way (see [14,20]). Since we consider a nonstationary problem, we further need the following assumption (see [9,22]), that is easily checked in an adaptive context:…”

Section: Lower Error Boundmentioning

confidence: 99%

“…To be able to extend these techniques to nonconforming spatial approximations, as for elliptic problems [14], we need to be able to estimate the consistency term appearing in the error equation. As in [14], this term is managed using a Helmholtz decomposition of the error. This allows us to extend the results from [7][8][9]22] to the nonconforming case.…”

Section: Introductionmentioning

confidence: 99%

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A posteriorierror estimates for a nonconforming finite element discretization of the heat equation

Nicaise¹,

Soualem²

2005

ESAIM: M2AN

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Abstract. The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in R d , d = 2 or 3, using backward Euler's scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main results with minimal assumptions on the mesh. Numerical experiments and a space-time adaptive algorithm confirm the theoretical predictions.Mathematics Subject Classification. 65N15, 65N30, 65M50.

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Section: Lower Error Boundmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A posteriorierror estimates for a nonconforming finite element discretization of the heat equation

Nicaise¹,

Soualem²

2005

ESAIM: M2AN

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“…Observe that this also provides L 2 (Ω) error control because of the discrete Poincaré-Friedrichs inequality. To estimate the error we shall make use of the following lemma, which is proved in [11] and used in [4,9].…”

Section: 7mentioning

confidence: 99%

“…Various alternative a posteriori estimators for the error in the broken energy norm have been proposed for the lowest order nonconforming Crouzeix-Raviart finite element approximation on triangular elements, including explicit residualbased estimators [11], hierarchical basis estimators [14] and averaging type estimators [10]. All of these estimators provide two-sided bounds on the error up to unknown positive constants that are nevertheless independent of the mesh-size.…”

Section: Introductionmentioning

confidence: 99%

Robust a posteriori error estimation for the nonconforming Fortin--Soulie finite element approximation

Ainsworth

Rankin

2008

Math. Comp.

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Abstract. We obtain a computable a posteriori error bound on the broken energy norm of the error in the Fortin-Soulie finite element approximation of a linear second order elliptic problem with variable permeability. This bound is shown to be efficient in the sense that it also provides a lower bound for the broken energy norm of the error up to a constant and higher order data oscillation terms. The estimator is completely free of unknown constants and provides a guaranteed numerical bound on the error.

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“…The first results proving the equivalence between the error and a residual type estimator were based on the use of a Helmholtz type decomposition of the error [9]. See also [3,7,11] where similar techniques were applied for mixed finite element approximations.…”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimates for non-conforming approximation of eigenvalue problems

Dari

Durán

Padra

2012

Applied Numerical Mathematics

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Abstract. We consider the approximation of eigenvalue problem for the laplacian by the Crouzeix-Raviart non conforming finite elements in two and three dimensions.Extending known techniques for source problems, we introduce a posteriori error estimators for eigenvectors and eigenvalues. We prove that the error estimator is equivalent to the energy norm of the eigenvector error up to higher order terms. Moreover, we prove that our estimator provides an upper bound for the error in the approximation of the first eigenvalue, also up to higher order terms.We present numerical examples of an adaptive procedure based on our error estimator in two and three dimensions. These examples show that the error in the adaptive procedure is optimal in terms of the number of degrees of freedom.

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A posteriori error estimators for nonconforming finite element methods

Cited by 127 publications

References 12 publications

A posteriorierror estimates for a nonconforming finite element discretization of the heat equation

A posteriorierror estimates for a nonconforming finite element discretization of the heat equation

Robust a posteriori error estimation for the nonconforming Fortin--Soulie finite element approximation

A posteriori error estimates for non-conforming approximation of eigenvalue problems

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