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

0425594 - MÚ 2015 RIV US eng J - Journal Article
Šebestová, I. - Vejchodský, Tomáš
Two-sided bounds for eigenvalues of differential operators with applications to Friedrichs', Poincaré, trace, and similar constants.
SIAM Journal on Numerical Analysis. Roč. 52, č. 1 (2014), s. 308-329. ISSN 0036-1429. E-ISSN 1095-7170
Institutional support: RVO:67985840
Keywords : bounds on spectrum * a posteriori error estimate * optimal constant
Subject RIV: BA - General Mathematics
Impact factor: 1.788, year: 2014
http://epubs.siam.org/doi/abs/10.1137/13091467X

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-Poincaré type. The abstract results are then applied to Friedrichs, Poincaré, and trace inequalities and fully computable two-sided bounds on the optimal constants in these inequalities are obtained. Accuracy of the method is illustrated in numerical examples.
Permanent Link: http://hdl.handle.net/11104/0231422