Two-Sided Bounds for Eigenvalues of Differential Operators with Applications to Friedrichs, Poincaré, 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--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.