Hierarchal triangular elements using orthogonal polynomials

Hierarchal 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 paper introduces a new hierarchal triangular element in which the basis functions are constructed from orthogonal polynomials—Jacobi polynomials. The resulting element is shown to be better conditioned than the earlier hierarchal element of Rossow and Katz.1 Recursive formulas allow the complete set of basis functions for an element to be efficiently evaluated at a given point. In addition, the formulas can be used to generate pre‐computed (universal) matrices. Examples are given of universal matrices, up to order 4, for the generalized Helmholtz equation. An electromagnetic problem involving a length of transmission line is used to show the usefulness of the new elements.