Generally, higher order methods refer to third order or higher [1]. Two general beliefs that have resulted in more widespread use of lower order methods in academia and especially in industry are [1]: 1) Higher order methods are more expensive; 2) They are not needed for engineering accuracies. Referring to the former, efficiency is a better measure than solution cost. To investigate a method’s efficiency both error convergence rate and solution cost scaling versus number of elements and element order should be well-understood. While the former generally has a simple power form, the solution cost scaling is much more complicated and highly dependent on particular formulation of a finite element method (FEM). For example as a result of the global coupling of conventional continuous FEMs,[2] reports that even for 0.1% relative accuracies—which is tenfold the commonly used 1% range in many engineering applications—linear elements are more efficient than higher order elements for realistic 3D applications.

We investigate when higher order methods are more efficient in the context of a highly advanced spacetime discontinuous Galerkin (SDG) finite element method [3] where a novel use of characteristic structure of the wave equation in discretization, yields a method with linear solution complexity. Some aspects of our study are: 1) Unlike continuous FEMs the very local solution nature of the SDG method may favor higher order elements. This clearly advocates the use of higher order elements for realistic 3D applications; 2) In many dynamic applications, eg wave propagation, temporal order is as important as spatial order. We comment on general challenges in achieving arbitrary high temporal orders—particularly when more efficient nonspatially uniform orders are demanded—and discuss how the SDG method gracefully addresses these issues; 3) We demonstrate multi-field formulations can in fact be more efficient than single-field formulations; 4) An FEM’s efficiency is greatly affected by the cost scaling of assembly and the solution of global matrices. We demonstrate this concept through comparison between single-cell and a new multi-cell element formulation.