A new error analysis for discontinuous finite element methods for linear elliptic problems

Gudi, Thirupathi

doi:10.1090/s0025-5718-10-02360-4

Cited by 189 publications

(167 citation statements)

References 33 publications

(41 reference statements)

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“…also [16,18], for example. An alternative proof of convergence for a range of (standard) DG methods, under minimal regularity assumptions, is presented by Gudi [13]; this is based on exploiting ideas from the a posteriori error analysis of DG methods. The analysis of the proposed DGCFEM in this setting will be investigated in the forthcoming article [12].…”

Section: A1429mentioning

confidence: 99%

$hp$-Version Composite Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

Antonietti¹,

Giani²,

Houston³

2013

SIAM J. Sci. Comput.

170

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Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. In this paper we introduce the hp-version discontinuous Galerkin composite finite element method for the discretization of second-order elliptic partial differential equations. This class of methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features, or microstructures. While standard numerical methods can be devised for such problems, the computational effort may be extremely high, as the minimal number of elements needed to represent the underlying domain can be very large. In contrast, the minimal dimension of the underlying composite finite element space is independent of the number of geometric features. The key idea in the construction of this latter class of methods is that the computational domain Ω is no longer resolved by the mesh; instead, the finite element basis (or shape) functions are adapted to the geometric details present in Ω. In this paper, we extend these ideas to the discontinuous Galerkin setting, based on employing the hp-version of the finite element method. Numerical experiments highlighting the practical application of the proposed numerical scheme will be presented.

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Section: A1429mentioning

confidence: 99%

$hp$-Version Composite Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

Antonietti¹,

Giani²,

Houston³

2013

SIAM J. Sci. Comput.

170

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“…The medius analysis of [15,25] proves for the discrete solution u CR ∈ CR (T) to (3.8) the best-approximation result…”

Section: Medius Analysismentioning

confidence: 91%

A New Generalization of the P ₁ Non-Conforming FEM to Higher Polynomial Degrees

Schedensack

2016

Computational Methods in Applied Mathematics

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This paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz spaces of arbitrary polynomial degree and its discretization coincides with the mentioned non-conforming FEM for the lowest polynomial degree. The discretization directly approximates the gradient of the solution instead of the solution itself. Besides the a priori and medius analysis, this paper proves optimal convergence rates for an adaptive algorithm for the new discretization. These are also demonstrated in numerical experiments. Furthermore, this paper focuses on extensions of this new scheme to quadrilateral meshes, mixed FEMs, and three space dimensions.

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“…We have followed the classical line for the error analysis to keep the presentation of the method simpler. Of course one might consider the extension of the results in [25] to estimate the nonconformity error arising in the nonconforming virtual approximation. We wish to note though, that such extension will require to have laid for virtual elements, some results on a-posteriori error estimation.…”

Section: Error Analysismentioning

confidence: 99%

The nonconforming virtual element method

2016

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Abstract. We introduce the nonconforming Virtual Element Method (VEM) for the approximation of second order elliptic problems. We present the construction of the new element in two and three dimensions, highlighting the main differences with the conforming VEM and the classical nonconforming finite element methods. We provide the error analysis and establish the equivalence with a family of mimetic finite difference methods.

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A new error analysis for discontinuous finite element methods for linear elliptic problems

Cited by 189 publications

References 33 publications

$hp$-Version Composite Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

$hp$-Version Composite Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

A New Generalization of the P ₁ Non-Conforming FEM to Higher Polynomial Degrees

The nonconforming virtual element method

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A new error analysis for discontinuous finite element methods for linear elliptic problems

Cited by 189 publications

References 33 publications

$hp$-Version Composite Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

$hp$-Version Composite Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

A New Generalization of the P 1 Non-Conforming FEM to Higher Polynomial Degrees

The nonconforming virtual element method

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A New Generalization of the P ₁ Non-Conforming FEM to Higher Polynomial Degrees