TY - JOUR
T1 - Analysis of a Cartesian PML Approximation to the Three Dimensional Electromagnetic Wave Scattering Problem
AU - Bramble , J. H.
AU - Pasciak , J. E.
JO - International Journal of Numerical Analysis and Modeling
VL - 3
SP - 543
EP - 561
PY - 2012
DA - 2012/09
SN - 9
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/646.html
KW - electromagnetic wave scattering problem, Maxwell scattering, Helmholtz equation, PML layer.
AB - <p style="text-align: justify;">We consider the application of a perfectly matched layer (PML) technique applied
in Cartesian geometry to approximate solutions of the electromagnetic wave (Maxwell) scattering
problem in the frequency domain. The PML is viewed as a complex coordinate shift (&quot;stretching&quot;)
and leads to a variable complex coefficient equation for the electric field posed on an infinite
domain, the complement of a bounded scatterer. The use of Cartesian geometry leads to a PML
operator with simple coefficients, although, still complex symmetric (non-Hermitian). The PML
reformulation results in a problem which preserves the original solution inside the PML layer
while decaying exponentially outside. The rapid decay of the PML solution suggests truncation
to a bounded domain with a convenient outer boundary condition and subsequent finite element
approximation (for the truncated problem).<br/>For suitably defined Cartesian PML layers, we prove existence and uniqueness of the solutions
to the infinite domain and truncated domain PML equations provided that the truncated domain
is sufficiently large. We show that the PML reformulation preserves the solution in the layer while
decaying exponentially outside of the layer. Our approach is to develop variational stability for
the infinite domain electromagnetic wave scattering PML problem from that for the acoustic wave
(Helmholtz) scattering PML problem given in [12]. The stability and exponential convergence of
the truncated PML problem is then proved using the decay properties of solutions of the infinite
domain problem. Although, we do not provide a complete analysis of the resulting finite element
approximation, we believe that such an analysis should be possible using the techniques in [6].</p>