Comparing DG and Nedelec finite element discretisations of the second-order time-domain Maxwell equation

This article compares the discontinuous Galerkin finite element method (DG-FEM) with the $H(\mathrm{curl})$-conforming FEM in the discretisation of the second-order time-domain Maxwell equations with possibly nonzero conductivity term. While DG-FEM suffers from an increased number of degrees of freedom compared with $H(\mathrm{curl})$-conforming FEM, it has the advantage of a purely block-diagonal mass matrix. This means that, as long as an explicit time-integration scheme is used, it is no longer necessary to solve a linear system at each time step -- a clear advantage over $H(\mathrm{curl})$-conforming FEM. It is known that DG-FEM generally favours high-order methods whereas $H(\mathrm{curl})$-conforming FEM is more suitable for low-order ones. The novelty we provide in this work is a direct comparison of the performance of the two methods when hierarchic $H(\mathrm{curl})$-conforming basis functions are used up to polynomial order $p=3$. The motivation behind this choice of basis functions is its growing importance in the development of $p$- and $hp$-adaptive FEMs. The fact that we allow for nonzero conductivity requires special attention with regards to the time-integration methods applied to the semi-discrete systems. High-order polynomial basis warrants the use of high-order time-integration schemes, but existing high-order schemes may suffer from a too severe time-step stability restriction as result of the conductivity term. We investigate several alternatives from the point of view of accuracy, stability and computational work. Finally, we carry out a numerical Fourier analysis to study the dispersion and issipation properties of the semi-discrete DG-FEM scheme and several of the time-integration methods. It is instructive in our approach that the dispersion and dissipation properties of the spatial discretisation and those of the time-integration methods are investigated separately, providing additional insight into the two discretisation steps.