Finite Element Transport Using Wachspress Rational Basis Functions on Quadrilaterals in Diffusive Regions

In 1975, Wachspress developed basis functions that can be constructed upon very general zone shapes, including convex polygons and polyhedra, as well as certain zone shapes with curved sides and faces. Additionally, Adams has recently shown that weight functions with certain properties will produce solutions with full resolution, meaning that they are capable of producing physically meaningful solutions in the diffusive limit. Wachspress rational functions (WRFs) possess these necessary properties. Here, we present methods to construct and integrate WRFs on quadrilaterals. We also present an asymptotic analysis of a discontinuous finite element discretization on quadrilaterals, and we present numerical results.