Guaranteed lower bounds for eigenvalues
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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.References
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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.
- Email: cc@mathematik.hu-berlin.de
- 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
- Email: jgedicke@math.lsu.edu
- 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.
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 83 (2014), 2605-2629
- MSC (2010): Primary 65N15, 65N25, 65N30
- DOI: https://doi.org/10.1090/S0025-5718-2014-02833-0
- MathSciNet review: 3246802