High-order Wachspress functions on convex polygons through computer algebra

The finite element method stands out as a powerful tool for modelling engineering problems. They are particularly well suited thanks to adaptive discretization techniques involving mesh size (h) or polynomial degree (p) or a combination of both (hp). In the case of p-adaptiveness, high-order function bases are required on the elements. For polygonal elements in 2D, Wachspress proved that, in the general case, it is not possible to construct a set of Lagrange finite elements with polynomial functions. Instead, this can be achieved through a popular set of basis functions which are generalized barycentric coordinates. One such family of functions is the Wachspress functions which are well suited for strictly convex polygons. While recent work has led to the construction of quadratic approximations from first-order Wachspress functions, there exists no approach for generalizing to higher orders for any strictly convex polygon. This work provides a general method to develop k-order Wachspress bases for any m-face polygon, where k