AN <i>A POSTERIORI</i> ERROR ESTIMATOR FOR hp-ADAPTIVE DISCONTINUOUS GALERKIN METHODS FOR ELLIPTIC EIGENVALUE PROBLEMS

Giani, Stefano; Hall, Edward

doi:10.1142/s0218202512500303

Cited by 33 publications

(46 citation statements)

References 30 publications

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“…The analysis has been inspired by [2] and [17]. The main difference between the a posteriori analysis for linear problems in [2] and the present a posteriori analysis for eigenvalue problems is the treatment of the higher order terms in the reliability results Theorem 4.2 and Corollary 4.6.…”

Section: A Posteriori Error Analysismentioning

confidence: 99%

hp-Adaptive composite discontinuous Galerkin methods for elliptic eigenvalue problems on complicated domains

Giani

2015

Applied Mathematics and Computation

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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-pro t 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. In this paper we develop the a posteriori error estimation of hpadaptive discontinuous Galerkin composite finite element methods (DGFEMs) for the discretization of second-order elliptic eigenvalue problems. DGFEMs allow for the approximation of problems posed on computational domains which may contain local geometric features. The dimension of the underlying composite finite element space is independent of the number of geometric features. This is in contrast with standard finite element methods, as the minimal number of elements needed to represent the underlying domain can be very large and so the dimension of the finite element space. Computable upper bounds on the error for both eigenvalues and eigenfunctions are derived. Numerical experiments highlighting the practical application of the proposed estimators within an automatic hp-adaptive refinement procedure will be presented.

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Section: A Posteriori Error Analysismentioning

confidence: 99%

hp-Adaptive composite discontinuous Galerkin methods for elliptic eigenvalue problems on complicated domains

Giani

2015

Applied Mathematics and Computation

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“…The anisotropic a posteriori error estimator is stated in Section 3 and a proof of its reliability given, up to higher order terms. The proof of reliability follows the same general idea as that presented in [37], which in turn followed from work in [14,9,12]. In Section 4 we present three numerical experiments to validate our theoretical results.…”

Section: Introductionmentioning

confidence: 84%

“…It is straightforward to prove efficiency of the error indicator (3.26) using the same techniques as in [37]; we omit the details for brevity. Unfortunately, as with many other works, for example [9,14,15], this efficiency result is robust only in terms of h. However, our numerical experiments indicate the error estimate to be robust in both h and p, even though theoretical results are not available.…”

Section: Remark 3 (Efficiency)mentioning

confidence: 99%

“…For example, they provide increased flexibility in mesh design (irregular grids are admissible) and the freedom to choose the elemental polynomial degrees without the need to enforce continuity between elements. Although a posteriori error analysis is a mature subject for source problems, for the approximation of eigenvalue problems relatively little work has been done; for the conforming FEM we refer the reader to [27,28,26,13] in the case of residual based error estimates and to [24] for a goal oriented approach; for a DG method, see our recent paper [37], where a robust residual error estimator is presented on isotropically refined grids, and [23,10] where the goal oriented approach is applied, the latter on anisotropic meshes. To the authors' knowledge, the work here represents a first attempt at residual based a posteriori error estimation for a DG method applied to an eigenvalue problem on anisotropic grids.…”

Section: Introductionmentioning

confidence: 99%

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An a-posteriori error estimate for $$hp$$ -adaptive DG methods for elliptic eigenvalue problems on anisotropically refined meshes

Giani

Hall

2013

Computing

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(2013) 'An a-posteriori error estimate for hp-adaptive DG methods for elliptic eigenvalue problems on anisotropically rened meshes.', Computing., 95 (1 Supplement). S319-S341.Further information on publisher's website:http://dx.doi.org/10.1007/s00607-012-0261-5Publisher's copyright statement:The nal publication is available at Springer via http://dx.doi.org/10.1007/s00607-012-0261-5Additional 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 We prove an a-posteriori error estimate for an hp-adaptive discontinuous Galerkin method for the numerical solution of elliptic eigenvalue problems with discontinuous coefficients on anisotropically refined rectangular elements. The estimate yields a global upper bound of the errors for both the eigenvalue and the eigenfunction and lower bound of the error for the eigenfunction only. The anisotropy of the underlying meshes is incorporated in the upper bound through an alignment measure. We present a series of numerical experiments to test the flexibility and robustness of this approach within a fully automated hp-adaptive refinement algorithm.

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“…For instance, Amara et al [2] developed the upper bound error estimates for the Helmholtz equation in planewave basis enriched DG method, and the error is measured in the L 2 -norm. Kaye et al [18] developed the upper bound error estimates for solving linear eigenvalue problems using non-polynomial basis functions in a DG framework, which generalizes the work of Giani et al [11] for polynomial basis functions. However, the assumption of approximation properties on the function space is in general difficult to verify.…”

Section: Previous Workmentioning

confidence: 99%

A posteriorierror estimates for discontinuous Galerkin methods using non-polynomial basis functions Part I: Second order linear PDE

Lin

Stamm

2016

ESAIM: M2AN

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Abstract. We present the first systematic work for deriving a posteriori error estimates for general non-polynomial basis functions in an interior penalty discontinuous Galerkin (DG) formulation for solving second order linear PDEs. Our residual type upper and lower bound error estimates measure the error in the energy norm. The main merit of our method is that the method is parameter-free, in the sense that all but one solution-dependent constants appearing in the upper and lower bound estimates are explicitly computable by solving local eigenvalue problems, and the only non-computable constant can be reasonably approximated by a computable one without affecting the overall effectiveness of the estimates in practice. As a side product of our formulation, the penalty parameter in the interior penalty formulation can be automatically determined as well. We develop an efficient numerical procedure to compute the error estimators. Numerical results for a variety of problems in 1D and 2D demonstrate that both the upper bound and lower bound are effective.

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AN A POSTERIORI ERROR ESTIMATOR FOR hp-ADAPTIVE DISCONTINUOUS GALERKIN METHODS FOR ELLIPTIC EIGENVALUE PROBLEMS

Cited by 33 publications

References 30 publications

hp-Adaptive composite discontinuous Galerkin methods for elliptic eigenvalue problems on complicated domains

hp-Adaptive composite discontinuous Galerkin methods for elliptic eigenvalue problems on complicated domains

An a-posteriori error estimate for $$hp$$ -adaptive DG methods for elliptic eigenvalue problems on anisotropically refined meshes

A posteriorierror estimates for discontinuous Galerkin methods using non-polynomial basis functions Part I: Second order linear PDE

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