engleski

On Estimators for Eigenvalue/Eigenvector Approximations

We consider a large class of residuum based \textit{; ; ; ; ; 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 a use of the strengthened Cauchy-Schwarz inequality and the saturation assumption, and gives reasonable and explicitly computable upper bounds on the discretization error. 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.

eigenvalue problems ; finite element methods ; a posteriori error estimation

An earlier version is available as a preprint from http://www.mis.mpg.de/publications/preprints/2007/prepr2007-18.html. A reader is advised to prefer the published version since it contains the most recent information.