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arXiv · 1303.7416

Two-sided bounds for eigenvalues of differential operators with applications to Friedrichs', Poincar\'e, trace, and similar constants

Ivana Šebestová·Tomáš Vejchodský
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Abstract

We present a general numerical method for computing guaranteed two-sided bounds for principal eigenvalues of symmetric linear elliptic differential operators. The approach is based on the Galerkin method, on the method of a priori-a posteriori inequalities, and on a complementarity technique. The two-sided bounds are formulated in a general Hilbert space setting and as a byproduct we prove an abstract inequality of Friedrichs'-Poincar\'e type. The abstract results are then applied to Friedrichs', Poincar\'e, and trace inequalities and fully computable two-sided bounds on the optimal constants in these inequalities are obtained. Accuracy of the method is illustrated on numerical examples.

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Ivana Šebestová, Tomáš Vejchodský. 2013-03-29. Two-sided bounds for eigenvalues of differential operators with applications to Friedrichs', Poincar\'e, trace, and similar constants. https://doi.org/10.1137/13091467x

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