COMPUTATIONAL LOWER BOUNDS OF THE MAXWELL

A method to compute guaranteed lower bounds to the eigenvalues of the Maxwell system in two or three space dimensions is proposed as a generalization of the method of Liu and Oishi [SIAM J. Numer. Anal., 51 (2013), pp. 1634--1654] for the Laplace operator. The main tool is the computation of an explicit upper bound to the error of the Galerkin projection. The error is split into two parts. One part is controlled by a hypercircle principle and an auxiliary eigenvalue problem. The second part requires a perturbation argument for the right-hand side replaced by a suitable piecewise polynomial. The latter error is controlled through the use of the commuting quasi-interpolation by Falk and Winther and computational bounds on its stability constant. This situation is different from the Laplace operator where such a perturbation is easily controlled through local Poincare'\ inequalities. The practical viability of the approach is demonstrated in test cases for two and three space dimensions.