A new triangular and tetrahedral basis for high‐order (<i>hp</i>) finite element methods

Abstract

Abstract In this paper we describe the foundations of a new hierarchical modal basis suitable for high‐order ( hp ) finite element discretizations on unstructured meshes. It is based on a generalized tensor product of mixed‐weight Jacobi polynomials. The generalized tensor product property leads to a low operation count with the use of sum factorization techniques. Variable p ‐order expansions in each element are readily implemented which is a crucial property for efficient adaptive discretizations. Numerical examples demonstrate the exponential convergence for smooth solutions and the ability of this formulation to handle easily very complex two‐ and three‐dmensional computational domains employing standard meshes.