Two-sided bounds for eigenvalues of differential operators with applications to Friedrichs', Poincaré, trace, and similar constants
For researchers in numerical analysis and PDEs, this provides a rigorous method to bound eigenvalue-related constants, though it is an incremental extension of existing techniques.
The paper presents a general numerical method for computing guaranteed two-sided bounds for principal eigenvalues of symmetric linear elliptic differential operators, achieving fully computable two-sided bounds on optimal constants in Friedrichs', Poincaré, and trace inequalities. Numerical examples illustrate the accuracy.
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 on numerical examples.