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From h to p Efficiently: Selecting the Optimal Spectral/hp Discretisation in Three Dimensions

Published online by Cambridge University Press:  16 May 2011

C. D. Cantwell ,
S. J. Sherwin ,
R. M. Kirby  and
P. H. J. Kelly
C. D. Cantwell*
Affiliation:
Department of Mathematics, Imperial College London, London, SW7 2AZ, UK
S. J. Sherwin
Affiliation:
Department of Aeronautics, Imperial College London, London, SW7 2AZ, UK
R. M. Kirby
Affiliation:
School of Computing, University of Utah, Salt Lake City, Utah, USA
P. H. J. Kelly
Affiliation:
Department of Computing, Imperial College London, London, SW7 2AZ, UK
*
⋆ Corresponding author. E-mail: c.cantwell@imperial.ac.uk
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Abstract

There is a growing interest in high-order finite and spectral/hp element methods using continuous and discontinuous Galerkin formulations. In this paper we investigate the effect of h- and p-type refinement on the relationship between runtime performance and solution accuracy. The broad spectrum of possible domain discretisations makes establishing a performance-optimal selection non-trivial. Through comparing the runtime of different implementations for evaluating operators over the space of discretisations with a desired solution tolerance, we demonstrate how the optimal discretisation and operator implementation may be selected for a specified problem. Furthermore, this demonstrates the need for codes to support both low- and high-order discretisations.

Keywords

elementoptimisationcode performance
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 3: Computational aerodynamics , 2011 , pp. 84 - 96
Copyright
© EDP Sciences, 2011

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References

Références

Bernard, P. E., Remacle, J. F., Comblen, R., Legat, V., Hillewaert, K.. High-order discontinuous Galerkin schemes on general 2D manifolds applied to the shallow water equations. J. Comput. Phys., 228 (2009), No. 17, 6514–6535. CrossRefGoogle Scholar
Cantwell, C. D., Sherwin, S. J., Kirby, R. M., Kelly, P. H. J.. From h to p efficiently: strategy selection for operator evaluation on hexahedral and tetrahedral elements. Computers and Fluids, 43 (2011), No. 1, 23–28. CrossRefGoogle Scholar
M. O. Deville, P. F. Fischer, E. H. Mund. High-order methods for incompressible fluid flow. Cambridge University Press, Cambridge, 2002.
Dubiner, M.. Spectral methods on triangles and other domains. J. Sci. Comp., 6 (1991), No. 4, 345-390. CrossRefGoogle Scholar
D. Gottlieb, S. A. Orszag. Numerical analysis of spectral methods: theory and applications. Society for Industrial Mathematics, 1977.
Hesthaven, J. S., Warburton, T.. Nodal high-order methods on unstructured grids:: I. time–domain solution of MaxwellŠs equations. J. Comput. Phys., 181 (2002), No. 1, 186-221. CrossRefGoogle Scholar
T. J. R. Hughes. The finite element method. Prentice-Hall, New Jersey, 1987.
G. E. Karniadakis and S. J. Sherwin. Spectral/hp element methods for computational fluid dynamics. Oxford University Press, Oxford, second edition edition, 2005.
U. Lee. Spectral element method in structural dynamics. Wiley, 2009.
S. A. Orszag. Spectral methods for problems in complex geometries. Advances in computer methods for partial differential equations- III, (1979), 148-157.
Patera, A. T.. A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys., 54 (1984), No. 3, 468-488. CrossRefGoogle Scholar
Sherwin, S. J., Karniadakis, G. E.. Tetrahedral hp finite elements: Algorithms and flow simulations. J. Comput. Phys., 124 (1996), 14-45. CrossRefGoogle Scholar
Sherwin, S. J.. Hierarchical hp finite elements in hybrid domains. Finite Elements in Analysis and Design, 27 (1997), No 1, 109-119. CrossRefGoogle Scholar
B. F. Smith, P. Bjorstad, W. Gropp. Domain decomposition: parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, 2004.
Vos, P. E. J., Sherwin, S. J., Kirby, M.. From h to p efficiently: Implementing finite and spectral/hp element discretisations to achieve optimal performance at low and high order approximations. J. Comput. Phys., 229 (2010), 5161-5181. CrossRefGoogle Scholar
O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu. The finite element method: its basis and fundamentals. Elsevier Butterworth Heinemann, 2005.