A Convergent Adaptive Method for Elliptic Eigenvalue Problems

Abstract

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...

Computation of eigenvalues by numerical upscaling

“…In cases with singular eigenfunctions (due to re-entrant corners in the domain or isolated jumps of the coefficient), one might as well use modern mesh-adaptive algorithms driven by some a posteriori error estimator as proposed and analyzed, e.g., in [Lar00], [Ney02], [DPR03], [CG11], [GMZ09], [GG09], [MM11], [CG12], [BGO13]. We are not competing with these efficient algorithms.…”

Convergent Adaptive Finite Element Methods for Photonic Crystal Applications

“…This formula comes from [20] where it has been shown to hold for standard elliptic problems with discontinuous coefficients and using a slightly different adaptive algorithm, which is not oscillation-free. For a sequence of computed eigenpairs on a sequence of adapted meshes either using the "standard" or the "modified" error estimator we estimate numerically the quantity γ using the formula: γ := |λ j − λ j,n |/|λ j − λ j,n−1 |.…”

“…The first proofs to appear had some extra assumptions on the initial mesh and some extra marking strategies to control the oscillations [19,20]. Then newer proofs appeared with no extra assumptions or oscillations strategies [7,17,18,9].…”

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

“…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.…”