TWO-SIDED BOUNDS FOR EIGENVALUES OF DIFFERENTIAL OPERATORS WITH APPLICATIONS TO FRIEDRICHS, POINCARE, TRACE, AND SIMILAR CONSTANTS

• ID: 2666, RIV: 10283100
• ISSN: 0036-1429, ISBN: not specified
• source: SIAM Journal on Numerical Analysis
• keywords: bounds on spectrum; a posteriori error estimate; optimal constant; Friedrichs inequality; Poincare inequality; trace inequality; Hilbert space
• authors: Ivana Šebestová, Tomáš Vejchodský
• authors from KNM: not assigned

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-Poincare type. The abstract results are then applied to Friedrichs, Poincare, 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.