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We present a general numerical method for computing guaranteed two-sidedbounds for principal eigenvalues of symmetric linear elliptic differentialoperators. The approach is based on the Galerkin method, on the method of apriori-a posteriori inequalities, and on a complementarity technique. Thetwo-sided bounds are formulated in a general Hilbert space setting and as abyproduct we prove an abstract inequality of Friedrichs'-Poincar\'e type. Theabstract results are then applied to Friedrichs', Poincar\'e, and traceinequalities and fully computable two-sided bounds on the optimal constants inthese inequalities are obtained. Accuracy of the method is illustrated onnumerical examples.

Two-sided bounds for eigenvalues of differential operators with applications to Friedrichs', Poincar\'e, trace, and similar constants