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Electromagnetic eigenfields in checkerboard patterns
Author(s) -
Kjellmert B.
Publication year - 2016
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22000
Subject(s) - mathematics , helmholtz equation , mathematical analysis , eigenfunction , helmholtz free energy , partial derivative , maxwell's equations , boundary value problem , geometry , eigenvalues and eigenvectors , physics , quantum mechanics
Here are described four solvers for time‐harmonic electromagnetic fields in checkerboard patterns. A pattern is built by four squares with constant permittivity,ε 1orε 2 . It is enclosed by conducting walls or is a unit cell of a periodic structure. The field is represented in two ways: by B , the transverse component of the magnetic induction, and by A , the magnetic vector potential in Lorenz gauge. B and A satisfy Helmholtz equations in each square as well as transmission and boundary conditions (BCs). These governing equations yield eigensolutions B and A , which are found to beH 1at worst. Variational versions of the governing equations are introduced. The weak formulations for B are standard, while those for A are new. They imply that the derivative transmission and BCs are satisfied weakly on interfaces between regions with different permittivity. Eigenpairs are computed approximately by spectral element methods. They yield mutually consistent eigenpairs. However, only about half of the eigenpairs ( ω 2 , A ) correspond to eigenpairs ( ω 2 , B ). For each set of BCs, the first few eigenfrequencies ω are given by tables, and some of the eigenfunctions are presented by contour plots. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 418–444, 2016