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**1 - 6**of**6**### VARIATIONAL MULTISCALE A POSTERIORI ERROR ESTIMATION FOR 2nd AND 4th-ORDER ODES

"... Abstract. In this paper, an explicit a posteriori error estimator is developed for second and fourth order ODEs solved with the Galerkin method that, remarkably, provides exact pointwise error estimates. The error estimator is derived from the variational multiscale theory, in which the subgrid scal ..."

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Abstract. In this paper, an explicit a posteriori error estimator is developed for second and fourth order ODEs solved with the Galerkin method that, remarkably, provides exact pointwise error estimates. The error estimator is derived from the variational multiscale theory, in which the subgrid scales are approximated making use of fine-scale Green’s functions. This methodology can be extended to any element type and order. Second and fourth order differential equations cover a great variety of problems in mechanics. Two examples with application in elasticity have been studied: the axially loaded beam and the Euler-Bernoulli beam. Because the error estimator is explicit, it can be very easily implemented and its computational cost is very small. Apart from pointwise error estimates, we present local and global a posteriori error estimates in the L1-norm, the L2-norm and the H1-seminorm. Finally, convergence rates of the error and the efficiencies of the estimator are analyzed.

### On

, 2000

"... a general decomposition of the error of an approximate stress field in elasticity ..."

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a general decomposition of the error of an approximate stress field in elasticity

### A Posteriori Error Estimator for Linear Elliptic Problem ∗

"... Abstract This paper concerns with a new error estimator for finite element approximation to the linear elliptic problem. A posteriori error estimator employing both a residual and a recovery based estimator is introduced. The error estimator is constructed by employing the recovery gradient method t ..."

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Abstract This paper concerns with a new error estimator for finite element approximation to the linear elliptic problem. A posteriori error estimator employing both a residual and a recovery based estimator is introduced. The error estimator is constructed by employing the recovery gradient method to obtain the approximated solutions of the linear elliptic problem. These solutions are combined with the residual method to produce the error estimator. Numerical results for selected test problems are demonstrated for the resulting error estimators and discussed.

### Error estimation for adaptive computations of shell structures

"... ABSTRACT. The finite element discretization of a shell structure introduces two kinds of errors: the error in the functional approximation and the error in the geometry approximation. The first is associated with the finite dimensional interpolation space and is present in any finite element computa ..."

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ABSTRACT. The finite element discretization of a shell structure introduces two kinds of errors: the error in the functional approximation and the error in the geometry approximation. The first is associated with the finite dimensional interpolation space and is present in any finite element computation. The latter is associated with the piecewise polynomial approximation of a curved surface and is much more relevant in shell problems than in any other standard 2D or 3D computation. In this work, a residual type error estimator introduced for standardfinite element analysis is generalized to shell problems. This allows easily to account for the real original geometry of the problem in the error estimation procedure and precludes the necessity of comparing generalized stress components between non coplanar elements. That is, the main drawbacks offlux projection error estimators are avoided. RESUME. La discretisation par elements finis d'une coque introduit deux types d'erreurs dif-ferentes: l'erreur dans l'approximation fonctionnelle et l'erreur dans l'approximation de la geometrie. La premiere est associee a l'espace d'interpolation qui est de dimension finie et apparaft dans n'importe quel calcul par elements finis. La deuxieme est liee a remplacer la sUrface courbe de la coque par un domaine polyedrique. Ce phinomene est beaucoup plus important dans les coques que dans les problemes standard 2-D ou 3-D. Cet article presente la generalisation aux elements de coques d'un estimateur d'erreur de type residuel qui avait ete introduit pour des elements finis standard. Cet estimateur permet de tenir compte dans Ie pro-cessus d'estimation de l'erreur de la geomitrie du probleme continu originel. II permet aussi d'eviter la comparaison des composantes des contraintes generalisees entre des elements non coplanaires, dont les estimateurs de projection de flux ne sa vent pas s ' en passer.

### upper bounds of the error

"... Equilibrated patch recovery error estimates: Simple and accurate upper bounds of the error ..."

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Equilibrated patch recovery error estimates: Simple and accurate upper bounds of the error