  The purpose of this research is to describe an efficient iterative method
suitable for obtaining high accuracy solutions to high frequency time-harmonic
scattering problems. The method allows for both refinement of local polynomial
degree and non-conforming mesh refinement, including multiple hanging nodes per
edge. Rather than globally assemble the finite element system, we describe an
iterative domain decomposition method which can use element-wise fast solvers
for elements of large degree. Since continuity between elements is enforced
through moment equations, the resulting constraint equations are hierarchical.
We show that, for high frequency problems, a subset of these constraints should
be directly enforced, providing the coarse space in the dual-primal domain
decomposition method. The subset of constraints is chosen based on a dispersion
criterion involving mesh size and wavenumber. By increasing the size of the
coarse space according to this criterion, the number of iterations in the
domain decomposition method depends only weakly on the wavenumber. We
demonstrate this convergence behaviour on standard domain decomposition test
problems and conclude the paper with application of the method to
electromagnetic problems in two dimensions. These examples include beam
steering by lenses and photonic crystal waveguides, as well as radar cross
section computation for dielectric, perfect electric conductor, and
electromagnetic cloak scatterers.
