Dual-order parallelogram finite elements for the axisymmetric vector Poisson equation

A new family of parallelogram finite elements, in which the order of approximation can be separately specified for the two pairs of opposing sides, is described. These elements are uniquely useful in problems involving long, narrow subregions and successfully complement high-order triangular elements for a wide range of applications. In addition to their use for narrow subregions they effectively fill up large, open spaces (a traditional use of rectangular elements) and uniquely allow different orders of approximation to be achieved in different regions without sacrificing the conformity of the solution. The assembly of these elements at run-time is accomplished without costly numerical integration by the use of pre-calculated universal element matrices. The element matrices are calculated exactly for one-dimensional elements through sixth order and are less expensive to generate and smaller than the corresponding matrices for triangular elements. Element matrices are given for approximation orders up to three and a sample problem is solved.