High-order Wachspress functions on convex polygons through computer algebra

Abstract

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<m. This method relies on symbolic computation for the assembly of a linear system whose numerical solution gives the values of the unknown coefficients of the Wachspress functions, and can be viewed as a generalization of Dasgupta's GADJ algorithm.This work originates from the development of a high-order Discontinuous Galerkin (DG) scheme for the solution of a linear transport equation over honeycomb meshes. As such, the regular hexagon is the strictly convex polygon taken as an example to describe the proposed method. We ensure that the functions obtained enforce the constraints required for their application in a finite element approach. Furthermore, we have successfully applied these functions to a finite element discretization of the Poisson equation and provided a test case for the linear transport equation on honeycomb meshes.In addition, further results are provided for an irregular pentagon to show the generality of the approach.