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Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems


Authors: Carsten Carstensen, Dietmar Gallistl and Mira Schedensack
Journal: Math. Comp. 84 (2015), 1061-1087
MSC (2010): Primary 65M12, 65M60, 65N25
DOI: https://doi.org/10.1090/S0025-5718-2014-02894-9
Published electronically: October 20, 2014
MathSciNet review: 3315500
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Abstract: The nonconforming approximation of eigenvalues is of high practical interest because it allows for guaranteed upper and lower eigenvalue bounds and for a convenient computation via a consistent diagonal mass matrix in 2D. The first main result is a comparison which states equivalence of the error of the nonconforming eigenvalue approximation with its best-approximation error and its error in a conforming computation on the same mesh. The second main result is optimality of an adaptive algorithm for the effective eigenvalue computation for the Laplace operator with optimal convergence rates in terms of the number of degrees of freedom relative to the concept of a nonlinear approximation class. The analysis includes an inexact algebraic eigenvalue computation on each level of the adaptive algorithm which requires an iterative algorithm and a controlled termination criterion. The analysis is carried out for the first eigenvalue in a Laplace eigenvalue model problem in 2D.


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Additional Information

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

Dietmar Gallistl
Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin,Germany

Mira Schedensack
Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin,Germany

DOI: https://doi.org/10.1090/S0025-5718-2014-02894-9
Received by editor(s): August 29, 2012
Received by editor(s) in revised form: September 6, 2013
Published electronically: October 20, 2014
Additional Notes: This work was supported by the DFG Research Center Matheon and the Berlin Mathematical School.
Article copyright: © Copyright 2014 American Mathematical Society