HIERARCHICAL 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 i n 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 result ing element is shown to be better conditioned than the earlier hierarc hal element of Rossow and Katz.(1) Recursive formulas allow the comple te set of basis functions for an element to be efficiently evaluated a t a given point. In addition, the formulas can be used to generate pre -computed (universal) matrices. Examples are given of universal matric es, up to order 4, for the generalized Helmholtz equation. An electrom agnetic problem involving a length of transmission line is used to sho w the usefulness of the new elements.