Tetrahedral spectral elements for CFD

Abstract

In this paper we describe the foundation of a new hierarchical basis suitable for spectral and h - p type finite elements in complex, three-dimensional domains. It is based on Jacobi polynomials of mixed weights and allows for a variable order in each element, which is a crucial property for efficient adaptive discretizations on unstructured meshes. In addition, it has the desirable property of an O(N ²) spectrum scaling for the convective operator as in tensorial spectral methods; this is important in explicit time-integration of the Navier-Stokes equations. The general Galerkin formulation for elliptic equations and a high-order splitting scheme used to solve the two-dimensional Navier-Stokes equations has been documented in [2]. Here we elaborate on the three-dimensional basis and demonstrate its properties through different numerical examples. The new algorithms have been implemented in the general code N, εkT αr, which represents the new generation of spectral element methods for unstructured meshes.