Journal of Hydrodynamics

, Volume 30, Issue 1, pp 1–22 | Cite as

Spectral/hp element methods: Recent developments, applications, and perspectives

  • Hui Xu
  • Chris D. Cantwell
  • Carlos Monteserin
  • Claes Eskilsson
  • Allan P. Engsig-Karup
  • Spencer J. Sherwin
Open Access
Review Article
  • 7 Downloads

Abstract

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate a C0 - continuous expansion. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed.

Keywords

High-precision spectral/hp elements continuous Galerkin method discontinuous Galerkin method implicit large eddy simulation 

Notes

Acknowledgments

The authors kindly thank the Executive Editorin-Chief Prof. Lian-di Zhou for the invitation to contribute this review article and Dr. Wei Zhang for his contribution to conducting the tip-vortex simulation. H.X. and S.J.S would like to acknowledge support under EPSRC (Grant No. EP/L000407/1).

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© The Authors 2018

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Hui Xu
    • 1
  • Chris D. Cantwell
    • 1
  • Carlos Monteserin
    • 2
  • Claes Eskilsson
    • 4
    • 5
  • Allan P. Engsig-Karup
    • 2
    • 3
  • Spencer J. Sherwin
    • 1
  1. 1.Department of AeronauticsImperial College LondonLondonUK
  2. 2.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkLyngbyDenmark
  3. 3.Center for Energy Resources Engineering (CERE)Technical University of DenmarkLyngbyDenmark
  4. 4.Department of Civil EngineeringAalborg UniversityAalborg ØDenmark
  5. 5.Division Safety and TransportResearch Institutes of Sweden (RISE)BoråsSweden

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