  In this paper, we present an efficient spectral-element method (SEM) for
solving general two-dimensional Helmholtz equations in anisotropic media, with
particular applications in accurate simulation of polygonal invisibility
cloaks, concentrators and circular rotators arisen from the field of
transformation electromagnetics (TE). In practice, we adopt a transparent
boundary condition (TBC) characterized by the Dirichlet-to-Neumann (DtN) map to
reduce wave propagation in an unbounded domain to a bounded domain. We then
introduce a semi-analytic technique to integrate the global TBC with local
curvilinear elements seamlessly, which is accomplished by using a novel
elemental mapping and analytic formulas for evaluating global Fourier
coefficients on spectral-element grids exactly.
  From the perspective of TE, an invisibility cloak is devised by a singular
coordinate transformation of Maxwell's equations that leads to anisotropic
materials coating the cloaked region to render any object inside invisible to
observers outside. An important issue resides in the imposition of appropriate
conditions at the outer boundary of the cloaked region, i.e., cloaking boundary
conditions (CBCs), in order to achieve perfect invisibility. Following the
spirit of [48], we propose new CBCs for polygonal invisibility cloaks from the
essential "pole" conditions related to singular transformations. This allows
for the decoupling of the governing equations of inside and outside the cloaked
regions. With this efficient spectral-element solver at our disposal, we can
study the interesting phenomena when some defects and lossy or dispersive media
are placed in the cloaking layer of an ideal polygonal cloak.
