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 on numerical examples.

Extended numerical experiments and minor corrections of the previous version. This version has been accepted for publication by SIAM J. Numer. Anal

References in corpus (1)

• Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods

Cited by in corpus (3)

• Localization Analysis of an Energy-Based Fourth-Order Gradient Plasticity Model
• Robust error bounds for finite element approximation of reaction-diffusion problems with non-constant reaction coefficient in arbitrary space dimension
• Fully computable a posteriori error bounds for eigenfunctions