Two-sided bounds for eigenvalues of differential operators with applications to Friedrichs', Poincar\'e, trace, and similar constants
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.
Keywords
Cite
@article{arxiv.1303.7416,
title = {Two-sided bounds for eigenvalues of differential operators with applications to Friedrichs', Poincar\'e, trace, and similar constants},
author = {Ivana Šebestová and Tomáš Vejchodský},
journal= {arXiv preprint arXiv:1303.7416},
year = {2014}
}
Comments
Extended numerical experiments and minor corrections of the previous version. This version has been accepted for publication by SIAM J. Numer. Anal