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Guaranteed lower bounds for eigenvalues

Authors: Carsten Carstensen and Joscha Gedicke
Journal: Math. Comp. 83 (2014), 2605-2629
MSC (2010): Primary 65N15, 65N25, 65N30
Published electronically: April 25, 2014
MathSciNet review: 3246802
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Abstract: This paper introduces fully computable two-sided bounds on the eigenvalues of the Laplace operator on arbitrarily coarse meshes based on some approximation of the corresponding eigenfunction in the nonconforming Crouzeix-Raviart finite element space plus some postprocessing. The efficiency of the guaranteed error bounds involves the global mesh-size and is proven for the large class of graded meshes. Numerical examples demonstrate the reliability of the guaranteed error control even with an inexact solve of the algebraic eigenvalue problem. This motivates an adaptive algorithm which monitors the discretisation error, the maximal mesh-size, and the algebraic eigenvalue error. The accuracy of the guaranteed eigenvalue bounds is surprisingly high with efficiency indices as small as 1.4.

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Additional Information

Carsten Carstensen
Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany – and — Department of Computational Science and Engineering, Yonsei University, 120–749 Seoul, Korea.

Joscha Gedicke
Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
Address at time of publication: Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803

Keywords: Eigenvalue, adaptive finite element method, bounds
Received by editor(s): January 11, 2012
Received by editor(s) in revised form: October 28, 2012, and March 12, 2013
Published electronically: April 25, 2014
Additional Notes: This paper was supported by the DFG Research Center MATHEON “Mathematics for key technologies”, and the graduate school BMS “Berlin Mathematical School” in Berlin, and the World Class University (WCU) program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology R31-2008-000-10049-0.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.