TWO-SIDED BOUNDS FOR EIGENVALUES OF DIFFERENTIAL OPERATORS WITH APPLICATIONS TO FRIEDRICHS, POINCARE, 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-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.

Keywords

• bounds on spectrum
• a posteriori error estimate
• optimal constant
• Friedrichs inequality
• Poincare inequality
• trace inequality
• Hilbert space