An a posteriori estimator of eigenvalue/eigenvector error for penalty-type discontinuous Galerkin methods

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An a posteriori estimator of eigenvalue/eigenvector error for penalty-type discontinuous Galerkin methods. / Giani, Stefano; Grubisic, Luka; Hakula, Harri; Ovall, Jeffrey S.

In: Applied Mathematics and Computation, Vol. 319, 02.2018, p. 562-574.

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@article{90ddf9b4a6cb463caa3a3e787769bc05,
title = "An a posteriori estimator of eigenvalue/eigenvector error for penalty-type discontinuous Galerkin methods",
abstract = "We provide an abstract framework for analyzing discretization error for eigenvalue problems discretized by discontinuous Galerkin methods such as the local discontinuous Galerkin method and symmetric interior penalty discontinuous Galerkin method. The analysis applies to clusters of eigenvalues that may include degenerate eigenvalues. We use asymptotic perturbation theory for linear operators to analyze the dependence of eigenvalues and eigenspaces on the penalty parameter. We first formulate the DG method in the framework of quadratic forms and construct a companion infinite dimensional eigenvalue problem. With the use of the companion problem, the eigenvalue/vector error is estimated as a sum of two components. The first component can be viewed as a “non-conformity” error that we argue can be neglected in practical estimates by properly choosing the penalty parameter. The second component is estimated a posteriori using auxiliary subspace techniques, and this constitutes the practical estimate.",
keywords = "Eigenvalue problem, Finite element method, A posteriori error estimates, Discontinuous Galerkin method",
author = "Stefano Giani and Luka Grubisic and Harri Hakula and Ovall, {Jeffrey S.}",
year = "2018",
month = "2",
doi = "10.1016/j.amc.2017.07.007",
language = "English",
volume = "319",
pages = "562--574",
journal = "Applied Mathematics and Computation",
issn = "0096-3003",
publisher = "Elsevier Inc.",

}

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TY - JOUR

T1 - An a posteriori estimator of eigenvalue/eigenvector error for penalty-type discontinuous Galerkin methods

AU - Giani, Stefano

AU - Grubisic, Luka

AU - Hakula, Harri

AU - Ovall, Jeffrey S.

PY - 2018/2

Y1 - 2018/2

N2 - We provide an abstract framework for analyzing discretization error for eigenvalue problems discretized by discontinuous Galerkin methods such as the local discontinuous Galerkin method and symmetric interior penalty discontinuous Galerkin method. The analysis applies to clusters of eigenvalues that may include degenerate eigenvalues. We use asymptotic perturbation theory for linear operators to analyze the dependence of eigenvalues and eigenspaces on the penalty parameter. We first formulate the DG method in the framework of quadratic forms and construct a companion infinite dimensional eigenvalue problem. With the use of the companion problem, the eigenvalue/vector error is estimated as a sum of two components. The first component can be viewed as a “non-conformity” error that we argue can be neglected in practical estimates by properly choosing the penalty parameter. The second component is estimated a posteriori using auxiliary subspace techniques, and this constitutes the practical estimate.

AB - We provide an abstract framework for analyzing discretization error for eigenvalue problems discretized by discontinuous Galerkin methods such as the local discontinuous Galerkin method and symmetric interior penalty discontinuous Galerkin method. The analysis applies to clusters of eigenvalues that may include degenerate eigenvalues. We use asymptotic perturbation theory for linear operators to analyze the dependence of eigenvalues and eigenspaces on the penalty parameter. We first formulate the DG method in the framework of quadratic forms and construct a companion infinite dimensional eigenvalue problem. With the use of the companion problem, the eigenvalue/vector error is estimated as a sum of two components. The first component can be viewed as a “non-conformity” error that we argue can be neglected in practical estimates by properly choosing the penalty parameter. The second component is estimated a posteriori using auxiliary subspace techniques, and this constitutes the practical estimate.

KW - Eigenvalue problem

KW - Finite element method

KW - A posteriori error estimates

KW - Discontinuous Galerkin method

U2 - 10.1016/j.amc.2017.07.007

DO - 10.1016/j.amc.2017.07.007

M3 - Article

VL - 319

SP - 562

EP - 574

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

ER -

ID: 16129819