From h to p efficiently: Implementing finite and spectral/hp element methods to achieve optimal performance for low- and high-order discretisations

  • Authors:

  • Peter E. J. Vos;Spencer J. Sherwin;Robert M. Kirby

  • Affiliations:

  • Department of Aeronautics, Imperial College London, London, UK and Flemish Institute for Technological Research (Vito), Mol, Belgium;Department of Aeronautics, Imperial College London, London, UK;School of Computing, University of Utah, Salt Lake City, UT, USA

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2010

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Abstract

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 efficiently 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 efficient 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.