Computational lower bounds of the Maxwell eigenvalues

Abstract

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] 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 in 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--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 Poincar\'e inequalities. The practical viability of the approach is demonstrated in test cases for two and three space dimensions.