Residual<i>a posteriori</i>error estimation for the Virtual Element Method for elliptic problems

Veiga, L. Beirão da; Manzini, Gianmarco

doi:10.1051/m2an/2014047

Cited by 107 publications

(86 citation statements)

References 43 publications

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“…The term ρ p K is related to the oscillation of f , the datum of the problem (1), and is typical also in the finite element framework. On the other hand, the term ζ p K deals with the nonexactness of the discrete bilinear form and is standard in a posteriori error analysis of VEM, see [17,26].…”

Section: Discussionmentioning

confidence: 99%

A posteriori error estimation and adaptivity in hp virtual elements

2019

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An explicit and computable error estimator for the hp version of the virtual element method (VEM), together with lower and upper bounds with respect to the exact energy error, is presented. Such error estimator is employed to provide, following the approach of [42], hp adaptive mesh refinements for very general polygonal meshes. In addition, a novel VEM hp Clément quasi-interpolant, instrumental for the a posteriori error analysis, is introduced. The performances of the adaptive method are validated by a number of numerical experiments.

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Section: Discussionmentioning

confidence: 99%

A posteriori error estimation and adaptivity in hp virtual elements

2019

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“…Proof. We start the following chain of developments from the definition of the conformity error given in (12), note that [[ T u ]] = 0 on every mesh side and that the moments up to order k − 1 of [[ T h u ]] across all mesh interfaces are zero, and apply the Cauchy-Schwarz inequality in the last two steps:…”

Section: Remark 62mentioning

confidence: 99%

The nonconforming Virtual Element Method for eigenvalue problems

2019

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We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allow to treat in the same formulation the two-and three-dimensional case. We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of the L 2 -inner product, and we study the convergence properties of the corresponding discrete eigenvalue problem. The proposed schemes provide a correct approximation of the spectrum, in particular we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.

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“…For a posteriori error estimators, the ideal case we expect is the so-called asymptotic exactness. 3) is asymptotically exact for the linear virtual element method, which distinguishes it from the residual-type a posteriori error estimators for virtual element methods in the literature [12,13,18].…”

Section: 1mentioning

confidence: 99%

“…For virtual element methods, there are only a few work concerning on the a posteriori error estimation and adaptive algorithms. In [12], Beirão Da Veiga and Manzini derived a posteriori error estimators for C 1 virtual element methods. In [18], Cangiani et al proposed a posteriori error estimators for the C 0 conforming virtual element methods for solving second order general elliptic equations.…”

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confidence: 99%

Superconvergent gradient recovery for virtual element methods

Guo

Xie

Zhao

2019

Math. Models Methods Appl. Sci.

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Virtual element method is a new promising finite element method using general polygonal meshes. Its optimal a priori error estimates are well established in literature. In this paper, we take a different viewpoint. We try to uncover the superconvergent property of virtual element methods by doing some local post-processing only on the degrees of freedom. Using the linear virtual element method as an example, we propose a universal gradient recovery procedure to improve the accuracy of gradient approximation for numerical methods using general polygonal meshes. Its capability of serving as a posteriori error estimators in adaptive computation is also investigated. Compared to the existing residual-type a posteriori error estimators for the virtual element methods, the recovery-type a posteriori error estimator based on the proposed gradient recovery technique is much simpler in implementation and it is asymptotically exact. A series of benchmark tests are presented to numerically illustrate the superconvergence of recovered gradient and validate the asymptotic exactness of the recovery-based a posteriori error estimator.

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Residuala posteriorierror estimation for the Virtual Element Method for elliptic problems

Cited by 107 publications

References 43 publications

A posteriori error estimation and adaptivity in hp virtual elements

A posteriori error estimation and adaptivity in hp virtual elements

The nonconforming Virtual Element Method for eigenvalue problems

Superconvergent gradient recovery for virtual element methods

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