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

TL;DR

This paper reviews the spectral/hp element method, highlighting its formulation, applications in fluid dynamics and ocean engineering, and discussing challenges for broader scientific and engineering use.

Contribution

It provides a comprehensive overview of recent developments, applications, and future perspectives of the spectral/hp element method in computational science.

Findings

High-precision solutions with exponential error reduction

Successful application in fluid dynamics and ocean engineering

Discussion of challenges for complex applications

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 C0-continuous expansions. 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 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.