A posteriori estimators for obstacle problems by the hypercircle method

Braess, Dietrich; Hoppe, Ronald H. W.; Schöberl, Joachim

doi:10.1007/s00791-008-0104-2

Cited by 32 publications

(22 citation statements)

References 15 publications

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“…After the pioneering works [13] by Hlaváček, Haslinger, Nečas, and Lovíšek (see Theorem 4.2 in this book) and [1] by Ainsworth, Oden, and Lee, a huge amount of work has been performed on the a posteriori analysis of variational inequalities, see, e.g., [2], [5], [11], [14], [16], [19], [20], and the references therein. It can be noted that, in [11], a mixed problem coupling a variational equality and an inequality is also considered.…”

Section: Introductionmentioning

confidence: 99%

On the unilateral contact between membranes. Part 2: a posteriori analysis and numerical experiments

Belgacem¹,

Bernardi

Blouza³

et al. 2011

IMA Journal of Numerical Analysis

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Abstract:The contact between two membranes can be described by a system of variational inequalities, where the unknowns are the displacements of the membranes and the action of a membrane on the other one. A discretization of this system is proposed in Part 1 of this work, where the displacements are approximated by standard finite elements and the action by a local postprocessing which admits an equivalent mixed reformulation. Here, we perform the a posteriori analysis of this discretization and prove optimal error estimates. Next, we present numerical experiments that confirm the efficiency of the error indicators.Résumé: Le contact entre deux membranes peutêtre décrit par un système d'inéquations variationelles, où les inconnues sont les déplacements des deux membranes et l'action d'une membrane sur l'autre. Une discrétisation de ce système est présentée dans la première partie de ce travail, où les déplacements sont approchés par deséléments finis usuels et l'action par un post-traitement local qui admet une reformulation mixteéquivalente. Nous effectuons ici l'analyse a posteriori de cette discrétisation, qui mèneà des estimations d'erreur optimales. Puis nous présentons des expériences numériques qui confirment l'efficacité des indicateurs d'erreur.

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Section: Introductionmentioning

confidence: 99%

On the unilateral contact between membranes. Part 2: a posteriori analysis and numerical experiments

Belgacem¹,

Bernardi

Blouza³

et al. 2011

IMA Journal of Numerical Analysis

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“…In particular, the sharpness of an upper bound cannot be verified through the continuous dependence of the usual dual residual norm on the approximate solution; cf. [3,5,6,16,19], where averaging or residual techniques are considered. Insensitivity of estimators with respect to certain perturbations of the load f has been established by means of the notion of full-contact introduced in [11].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Hierarchical error estimates for the energy functional in obstacle problems

et al. 2011

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Abstract. We present a hierarchical a posteriori error analysis for the minimum value of the energy functional in symmetric obstacle problems. The main result is that the energy of the exact solution is, up to oscillation terms, equivalent to an appropriate hierarchical estimator. The proof does not invoke any saturation assumption. Moreover, we prove an a posteriori error estimate indicating that the estimator from [12] is asymptotically reliable and we give sufficient conditions for the validity of a saturation assumption. Finally, we corroborate and complement our theoretical results with a numerical example.

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“…The two-energies principle was originally established by Prager and Synge [30,35] for elliptic equations of second order under the name hypercircle method. It has been used by many authors, e.g., in [1,8,9,11,20,33] for the evaluation of a posteriori error estimates. The principle was reformulated several times in order to obtain error estimates by a postprocessing also when nonconforming finite elements are involved.…”

Section: A Two-energies Principle For the Biharmonic Equationmentioning

confidence: 99%

“…Error estimation using the two-energies principle. The dominating part of the overall discretization error will be estimated by using the two-energies principle (11). To this end, an equilibrated moment tensor σ eq h will be constructed.…”

Section: 2mentioning

confidence: 99%

An equilibration-based a posteriori error bound for the biharmonic equation and two finite element methods

Braess

Pechstein

Schöberl

2019

IMA Journal of Numerical Analysis

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We develop an a posteriori error estimator for the Interior Penalty Discontinuous Galerkin approximation of the biharmonic equation with continuous finite elements. The error bound is based on the two-energies principle and requires the computation of an equilibrated moment tensor. The natural space for the moment tensor consists of symmetric tensor fields with continuous normal-normal components. It is known from the Hellan-Herrmann-Johnson (HHJ) mixed formulation. We propose a construction that is totally local. The procedure can also be applied to the original HHJ formulation, which directly provides an equilibrated moment tensor.

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A posteriori estimators for obstacle problems by the hypercircle method

Cited by 32 publications

References 15 publications

On the unilateral contact between membranes. Part 2: a posteriori analysis and numerical experiments

On the unilateral contact between membranes. Part 2: a posteriori analysis and numerical experiments

Hierarchical error estimates for the energy functional in obstacle problems

An equilibration-based a posteriori error bound for the biharmonic equation and two finite element methods

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