We present a high order, Fourier penalty method for the Maxwell's equations
in the vicinity of perfect electric conductor boundary conditions. The approach
relies on extending the smooth non-periodic domain of the equations to a
periodic domain by removing the exact boundary conditions and introducing an
analytic forcing term in the extended domain. The forcing, or penalty term is
chosen to systematically enforce the boundary conditions to high order in the
penalty parameter, which then allows for higher order numerical methods. We
present an efficient numerical method for constructing the penalty term, and
discretize the resulting equations using a Fourier spectral method. We
demonstrate convergence orders of up to 3.5 for the one-dimensional Maxwell's
equations, and show that the numerical method does not suffer from dispersion
(or pollution) errors. We also illustrate the approach in two dimensions and
demonstrate convergence orders of 2.5 for transverse magnetic modes and 1.5 for
the transverse electric modes. We conclude the paper with numerous test cases
in dimensions two and three including waves traveling in an irregular
waveguide, and scattering off of a windmill-like geometry.Comment: 50 pages, 21 figures; added three-dimensional results, added
reference
