Polygonal Spline Spaces and the Numerical Solution of the Poisson Equation

Abstract

It is known that generalized barycentric coordinates can be used to form Bernstein polynomial-like functions over a polygon with any number of sides. We propose to use these functions to form a space of continuous polygonal splines (piecewise defined functions) of order $d$ over a partition consisting of polygons which is able to reproduce all polynomials of degree $d$. Locally supported basis functions for the space are constructed for order $d\ge 2$. The construction for $d=2$ is simpler than the “serendipity” quadratic finite elements that have appeared in the recent literature. The number of basis functions is similar to, but fewer than, those of the virtual element method. We use them for the numerical solution of the Poisson equation on two special types of nontriangular partitions to present a proof of concept for solving PDEs over polygonal partitions. Numerical solutions based on quadrangulations and pentagonal partitions are demonstrated to show the efficiency of these polygonal spline functions. They can lead to a more accurate solution by using fewer degrees of freedom than the traditional continuous polynomial finite element method if the solutions are smooth although assembling the mass and stiffness matrices can take more time.