Computational lower bounds of the Maxwell eigenvalues

TL;DR

This paper introduces a method to compute guaranteed lower bounds for Maxwell eigenvalues in 2D and 3D, extending previous techniques for the Laplace operator by using error bounds and perturbation analysis.

Contribution

It generalizes the Liu and Oishi method to Maxwell eigenvalues, providing a practical approach with explicit error bounds and stability analysis.

Findings

Method successfully computes lower bounds in test cases.

Approach is applicable to both 2D and 3D Maxwell problems.

Demonstrates practical viability through numerical experiments.

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.