Quadratic serendipity discontinuous finite element discretization for SN transport on arbitrary polygonal grids

In this paper, we present a quadratic serendipity discontinuous Galerkin finite element (DGFEM) discretization of the SN transport equation for arbitrary polygonal grids. The quadratic serendipity functions are constructed from products of linear Generalized Barycentric Coordinates (GBC) and are fully compatible with arbitrary polygonal grids. The piecewise linear (PWL) functions are a GBC that have been previously utilized for DGFEM transport. We employ these PWL functions, and others for comparison, as the underlying linear functions in this work. Once constructed, the quadratic serendipity functions span the {1,x,y,x2,xy,y2} space of functions and grow by 2n on a mesh element where n is the number of the polygon's vertices. Numerical tests confirm that the functions capture exactly quadratic solutions and appropriate convergence rates are observed, including a test case involving spatial adaptive mesh refinement. Finally, the functions are analyzed for diffusive problems and retain full resolution in the thick diffusion limit.