On estimators for eigenvalue/eigenvector approximations

Abstract

Abstract. We consider a large class of residuum based a posteriori eigenvalue/eigenvector estimates and present an abstract framework for proving their asymptotic exactness. Equivalence of the estimator and the error is also established. To demonstrate the strength of our abstract approach we present a detailed study of hierarchical error estimators for Laplace eigenvalue problems in planar polygonal regions. To this end we develop new error analysis for the Galerkin approximation which avoids the use of the strengthened CauchySchwarz inequality and the saturation assumption, and gives reasonable and explicitly computable upper bounds on the discretization error. A brief discussion is also given concerning the design of estimators which are in the same spirit, but are based on different a posteriori techniques-notably, those of gradient recovery type.

Benchmark results for testing adaptive finite element eigenvalue procedures part 2 (conforming eigenvector and eigenvalue estimates)

“…In Theorems 2.2 and 2.4 below, we state key theorems from [20] and its published version [17], which were used for finite element computations in [21,7]. The results of [20,17] show that these approximation defects would yield ideal error estimates for eigenvalue and eigenvector computation if they could be computed.…”

“…The constant C m is given by an explicit formula which is a reasonable practical overestimate, see [21,7,20,17,16] for details. In particular the requirement ηm(Ŝm) 1−ηm(Ŝm) < λ m+1 −λm λ m+1 +λm is according to [16,17] [20,17,16,18] in the finite element setting.…”

“…As in Part 1, we provide extensive numerical tests for comparison with other high-order methods. Our CG work builds on the abstract framework of [21,7] for eigenvalue/invariant subspace error estimation, as outlined in Section 2, providing a more robust theory than was possible in our previous approach. Concerning hp-adaptive approximation methods for eigenvalue problems, much more has been written about their a priori error analysis than their a posteriori error analysis, and we mention [5,26,31], as well as the more comprehensive general survey [11], as recent references in this vein.…”

“…In Section 2 we introduce basic notation related to the hp-discretization, and discuss approximation defects as ideal error estimates, with key results from [21,7] extended for use in the present context. These extensions make possible the incorporation of results from [28,Section 3] to obtain efficient and reliable estimates of eigenvalue and eigenvector approximations.…”

Guaranteed and Robust a Posteriori Bounds for Laplace Eigenvalues and Eigenvectors: Conforming Approximations

“…For conforming finite elements, relying on the a priori error estimates resumed in Babuška and Osborn [2] and Boffi [7], see also the references therein, a posteriori error estimates have been obtained by Verfürth [60], Maday and Patera [43], Larson [38], Heuveline and Rannacher [31], Durán et al [20], Grubišić and Ovall [29], Rannacher et al [50], andŠolín and Giani [56], see also the references therein. These estimates, though, systematically contain uncomputable terms, typically higher order on fine enough meshes.…”