Stability of Perfectly Matched Layers for Maxwell's Equations in Rectangular Solids
Résumé
Perfectly matched layers
are extensively used
to compute approximate solutions
for Maxwell's equations
in $\RR^{1+3}$
using a bounded computational domain,
usually a rectangular solid.
A smaller rectangular
domain of interest is surrounded by layers designed to absorb outgoing
waves in perfectly reflectionless manner. On the
boundary of the computational domain, an
absorbing boundary condition is imposed that is necessarily imperfect.
The method replaces
the Maxwell equations by a larger system,
and introduces
absorption coefficients
positive in the layers.
Well posedness of the resulting initial boundary value problem
is proved here
for the first time.
The Laplace transform of
a resulting Helmholtz system is studied.
For positive real values of the transform variable
$\tau$,
the Helmholtz system has a unique
solution from a variational form that yields limited
regularity for rectangular domains.
When $\tau$ is not real
the
complex variational form
loses positivity.
We smooth the domain and, in spite of this loss, construct $H^2$ solutions with uniform
$L^2$ estimates.
Using the $H^2$ regularity, we deduce Maxwell from Helmholtz,
then
remove the smoothing.
The boundary condition
at the smoothed boundary
must be carefully chosen.
A
method of Jerison-Kenig-Mitrea
is extended to
compensate
the nonpositivity of the flux.
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