Constructing Sibson Elements for a Rectangular Mesh

摘要

This paper documents the construction of a finite element, called the Sibson element. The shape function of this element is formed on rectangular grids by C(sub 1) splines defined on a triangulation of each subrectangle by dividing it into four subtriangles formed by drawing the diagonals. The splines are constructed from bivariate cubic polynomials z(x,y) and are written in such a way that they are linear functions of the values z, delta(z)/delta(x), delta(z)/delta(y) at each node of the rectangle with bivariate polynomial coefficients up to order three. Conditions are given for the existence of such an element. They are used to construct the bivariate polynomial coefficients, first for a unit rectangle and then for a general rectangle. Since the first and second derivatives of these functions are sometimes needed they are also given.