Explicit a posteriori error estimates for eigenvalue analysis of heterogeneous elastic structures

Walsh, Timothy; Reese, Garth M.; Hetmaniuk, Ulrich

doi:10.1016/j.cma.2006.10.036

Cited by 26 publications

(19 citation statements)

References 22 publications

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“…As with any numerical approach, it is important to be able to quantify the error made by way of an a 1 posteriori error estimate, which can then also be used to drive an adaptive mesh/polynomial enrichment process. Although a posteriori error analysis is a mature subject for source problems, for eigenvalues it is still very much in its infancy; for the conforming FEM we refer the reader to [24,25,23] in the case of residual based error estimates and to [21] for a goal oriented approach; for the discontinuous Galerkin finite element method (DGFEM) see [20] where the goal oriented approach is applied in the context of linear stability analysis for the incompressible Navier-Stokes equations. To the authors' knowledge, the work here represents a first attempt at residual based a posteriori error estimation for a DGFEM applied to an eigenvalue problem.…”

Section: Introductionmentioning

confidence: 99%

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

Giani

Hall

2012

Math. Models Methods Appl. Sci.

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In this paper we present a residual-based a posteriori error estimator for hp-adaptive discontinuous Galerkin methods for elliptic eigenvalue problems. In particular, we use as a model problem the Laplace eigenvalue problem on bounded domains in ℝd, d = 2, 3, with homogeneous Dirichlet boundary conditions. Analogous error estimators can be easily obtained for more complicated elliptic eigenvalue problems. We prove the reliability and efficiency of the residual-based error estimator also for non-convex domains and use numerical experiments to show that, under an hp-adaptation strategy driven by the error estimator, exponential convergence can be achieved, even for non-smooth eigenfunctions.

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Section: Introductionmentioning

confidence: 99%

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

Giani

Hall

2012

Math. Models Methods Appl. Sci.

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“…Analogous eigenvalue estimates can be found in [9] (for the Laplace problem) and [25] (for linear elasticity) and related results are in [14].…”

Section: )mentioning

confidence: 86%

A Convergent Adaptive Method for Elliptic Eigenvalue Problems

Giani¹,

Graham²

2009

SIAM J. Numer. Anal.

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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.We prove the convergence of an adaptive linear finite element method for computing eigenvalues and eigenfunctions of second-order symmetric elliptic partial differential operators. The weak form is assumed to yield a bilinear form which is bounded and coercive in H 1 . Each step of the adaptive procedure refines elements in which a standard a posteriori error estimator is large and also refines elements in which the computed eigenfunction has high oscillation. The error analysis extends the theory of convergence of adaptive methods for linear elliptic source problems to elliptic eigenvalue problems, and in particular deals with various complications which arise essentially from the nonlinearity of the eigenvalue problem. Because of this nonlinearity, the convergence result holds under the assumption that the initial finite element mesh is sufficiently fine.Key words. second-order elliptic problems, eigenvalues, adaptive finite element methods, convergence AMS subject classifications. 65N12, 65N25, 65N30, 65N50 DOI. 10.1137/070697264 1. Introduction. In the last decades, mesh adaptivity has been widely used to improve the accuracy of numerical solutions to many scientific problems. The basic idea is to refine the mesh only where the error is high, with the aim of achieving an accurate solution using an optimal number of degrees of freedom. There is a large amount of numerical analysis literature on adaptivity, in particular on reliable and efficient a posteriori error estimates (e.g., [1]). Recently, the question of convergence of adaptive methods has received intensive interest and a number of convergence results for the adaptive solution of boundary value problems have appeared (e.g., [8,18,19,7,6,23]).We prove here the convergence of an adaptive linear finite element algorithm for computing eigenvalues and eigenvectors of scalar symmetric elliptic partial differential operators in bounded polygonal or polyhedral domains, subject to Dirichlet boundary data. Such problems arise in many applications, e.g., resonance problems, nuclear reactor criticality, and the modelling of photonic band gap materials, to name but three.Our refinement procedure is based on two locally defined quantities, firstly, a standard a posteriori error estimator and secondly a measure of the variability (or "oscillation") of the computed eigenfunction. (Measures of "data oscillation" appear in the theory of adaptivity for boundary value problems, e.g., [18]. In the eigen...

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“…Duran et al (1999Duran et al ( , 2003 and Larson (2000); also see Noël (2004) for the eigenvalue problem associated with the Schrödinger operator and Walsh et al (2007) for heterogeneous elastic structures.…”

Section: Introductionmentioning

confidence: 99%