Polygonal Finite Elements of Arbitrary Order

Abstract

Finite elements with the shape of arbitrary polygons have been previously described. Originally they were first-order, i.e., able to represent exactly all polynomials of degree 1 in the space coordinates, but recently polygonal finite elements (PFEs) up to order 3 have been reported. Here, we propose a general theory for generating PFEs of arbitrary order. They are hierarchical, allowing mixing of orders. Results for a wave problem and two magnetic field problems show the effectiveness of the elements up to order 5.