A Mimetic Spectral Element Implementation of the Maxwell Eigenvalue Problem

Abstract

Mimetic formulations, also known as structure-preserving methods, are numerical schemes that preserve fundamental properties of the continuous differential operators at a discrete level. Additionally, they are well-known for satisfying constraints such as conservation of mass or momentum.

In the present work, a Mimetic Spectral Element Method based on quadrilaterals is explored. As an introduction, the framework is first implemented and tested on the classical Poisson equation, the Hartmann Flow system and several eigenvalue problems for the Laplacian operator. Solutions are attained by direct/mixed formulations and the extension to multi-element approaches is dealt with using either gathering or connectivity matrices. Different boundary conditions and various geometries are utilized.

Afterwards, the Maxwell Eigenvalue problem for the electric field E with general material properties is tackled in an attempt to generate spurious-free solutions by incorporating the condition ∇·D = 0 into the discrete system. The formulation is further scrutinized on geometries with Betti number b₁ > 0 as to verify if the proposed scheme captures the physical zero eigenvalues.

In the end, a mixed formulation for the eigenproblem is proposed in which the curl-curl operator is separated. The approximation of the electrostatic field energy is then computed with this formulation and compared to the solution obtained with a direct method allowing to create an upper and a lower bound for this variable.