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

Hui Xu, Chris Cantwell, Carlos Monteserin, Claes Eskilsson, Allan Engsig-Karup, Spencer Sherwin

Research output: Contribution to journalReview articlepeer-review

85 Citations (Scopus)


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.
Original languageEnglish
JournalJournal of Hydrodynamics
Issue number1
Pages (from-to)1-22
Publication statusPublished - 2018


  • High-precision spectral/hp elements
  • Continuous Galerkin method
  • Discontinuous Galerkin method
  • Implicit large eddy simulation


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