Hierarchical tetrahedral elements using orthogonal polynomials

Tetrahedral finite elements are widely used in 3D electromagnetics. They are the simplest shape into which a 3D region can be broken, and are well-suited to automatic mesh generation. Hierarchical elements are finite elements which have the useful property that elements with different polynomial orders can be used together in the same mesh without causing discontinuities. This is highly desirable, because it permits polynomial order to be used to control the distribution of the degrees of freedom. This paper introduces a new hierarchical tetrahedral element in which the basis functions are constructed from orthogonal polynomials (Jacobi polynomials), allowing mixing of polynomial orders up to three. Explicit basis functions are given in addition to the description of the linear independence property. As was the case for regular elements the pre-calculation of universal matrices will yield faster and more accurate results. The derivation and the corresponding universal matrices for the new elements are also shown. The new elements are used to solve for the electrostatic potential in a 3D region (where there is no analytical solution).