From h to p Efficiently: Implementing finite and spectral/hp element discretisations to achieve optimal performance at low and high order approximations.

The spectral/hp element method can be considered as bridging the gap between the – traditionally low order – finite element method on one side and spectral methods on the other side. Consequently, a major challenge which arises in implementing the spectral/hp element methods is to design algorithms that perform efficient ly for both low- and high-order spectral/hp discretisations, as well as discretisations in the intermediate regime. In this paper, we explain how the judicious use of different implementation strategies can be employed to achieve high efficiency across a wide range of polynomial orders. Furthermore, based upon this effic ient implementation, we analyse which spectral/hp discretisation (which specific combination of mesh size h and polynomial order P ) minimises the computational cost to solve an elliptic problem up to a predefined level of accuracy. We investigate this question for a set of both smooth and non-smooth problems.