  We present numerical experiments for geophysics electromagnetic (EM) modeling
based upon high-order edge elements and supervised $h+p$ refinement approaches
on massively parallel computers. Our high-order $h+p$ refinement strategy is
based on and extends the PETGEM code. We focus on the performance study in
terms of accuracy, convergence rate, and computational effort to solve
real-life 3D setups based on synthetic and experimental data for energy
reservoir characterization. These test cases show variable resolution
discretization needs and realistic physical parameters. In general, our
numerical results are consistent theoretically. The use of $h-$adapted meshes
was efficient to achieve a certain accuracy level in the synthetic EM
responses. Regarding global $p-$refinement, $p=2$ exhibits the best
accuracy/performance trade-off. Selective $p$-refinement might offer a better
compromise between accuracy and computational cost. However, for $p-$refinement
at different entities, the best refinement scheme consists of using $p=3$ at
the volume level with $p=1$ at faces and edges. Thus, $p-$refinement can be
competitive if applied hierarchically. Nevertheless, we acknowledge that the
performance of our supervised $h+p$ refinement strategy depends on the input
model (e.g., conductivity, frequency, domain decomposition strategy, among
others). Whatever the chosen configuration, our numerical results provide an
in-depth understanding of EM modeling's pros and cons when supervised $h+p$
refinement schemes are applied.
