Approximation by finite element functions using local regularization

Clément, Philippe

doi:10.1051/m2an/197509r200771

Cited by 731 publications

(652 citation statements)

References 8 publications

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“…Quasi-interpolant operator. Next we introduce a quasi-interpolant operator which is similar to the Clément operator [7]. The properties of this interpolant are essential to carry out the error analysis of the CIP method; see [2,3,4].…”

Section: 2mentioning

confidence: 99%

Weighted error estimates of the continuous interior penalty method for singularly perturbed problems

Burman

Guzmán

Leykekhman

2008

IMA Journal of Numerical Analysis

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Abstract.In this paper we analyze local properties of the Continuous Interior Penalty (CIP) Method for a model convection-dominated singularly perturbed convection-diffusion problem. We show weighted a priori error estimates, where the weight function exponentially decays outside the subdomain of interest. This result shows that locally, the CIP method is comparable to the Streamline Diffusion (SD) or the Discontinuous Galerkin (DG) methods.

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Section: 2mentioning

confidence: 99%

Weighted error estimates of the continuous interior penalty method for singularly perturbed problems

Burman

Guzmán

Leykekhman

2008

IMA Journal of Numerical Analysis

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“…We consider the projection operator P C : L 2 (Ω) → S 1 (Ω; T k ) with respect to a discrete L 2 -norm due to Clément [13]. In contrast to the Lagrangian interpolation operator, this operator can be applied to discontinuous functions.…”

Section: The Residual Based Error Estimatormentioning

confidence: 99%

A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

Wohlmuth

Hoppe

1999

Math. Comp.

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Abstract. We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a residual based estimator, a hierarchical one, error estimators relying on the solution of local subproblems and on a superconvergence result, respectively. Finally, we examine the relationship between the presented error estimators and compare their local components.

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“…For ψ ∈ H 1 0 (Ω) 2 , we denote by ψ I ∈ H 1 0 (Ω) 2 a piecewise linear average interpolant as defined in [9,18], satisfying…”

Section: Proofsmentioning

confidence: 99%

Approximation of the vibration modes of a plate by Reissner-Mindlin equations

et al. 1999

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Abstract. This paper deals with the approximation of the vibration modes of a plate modelled by the Reissner-Mindlin equations. It is well known that, in order to avoid locking, some kind of reduced integration or mixed interpolation has to be used when solving these equations by finite element methods. In particular, one of the most widely used procedures is the mixed interpolation tensorial components, based on the family of elements called MITC. We use the lowest order method of this family.Applying a general approximation theory for spectral problems, we obtain optimal order error estimates for the eigenvectors and the eigenvalues. Under mild assumptions, these estimates are valid with constants independent of the plate thickness. The optimal double order for the eigenvalues is derived from a corresponding L 2 -estimate for a load problem which is proven here. This optimal order L 2 -estimate is of interest in itself.Finally, we present several numerical examples showing the very good behavior of the numerical procedure even in some cases not covered by our theory.

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Approximation by finite element functions using local regularization

Cited by 731 publications

References 8 publications

Weighted error estimates of the continuous interior penalty method for singularly perturbed problems

Weighted error estimates of the continuous interior penalty method for singularly perturbed problems

A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

Approximation of the vibration modes of a plate by Reissner-Mindlin equations

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