Energy-Norm-Based and Goal-Oriented Automatic $hp$ Adaptivity for Electromagnetics: Application to Waveguide Discontinuities

The finite-element method (FEM) enables the use of ldquoadaptedrdquo meshes. The simultaneous combination of
h
(local variations in element size) and
p
(local variations in the polynomial order of approximation) refinements, i.e.,
hp
-adaptivity, is the most powerful and flexible type of adaptivity. In this paper, two versions of a fully automatic
hp
-adaptive FEM for electromagnetics are presented. The first version is based on minimizing the energy-norm of the error. The second, namely the goal oriented strategy, is based on minimizing the error of a given (user-prescribed) quantity of interest. The adaptive strategy delivers exponential convergence rates for the error, even in the presence of singularities. The
hp
adaptivity is presented in the context of 2-D analysis of
H
-plane rectangular waveguide discontinuities. Stabilized variational formulations and
H
(curl) FEM discretizations in terms of quadrilaterals of variable order of approximation supporting anisotropy and
hanging

nodes
are used. Comparison of energy-norm and goal-oriented
hp
-adaptive strategies in the context of waveguiding problems is provided. Specifically, the scattering parameters of the discontinuity are used as goal.