We analyze a numerical algorithm for solving radiative transport equation with vacuum or reflection boundary condition that was proposed in [4] with angular discretization by finite element method and spatial discretization by discontinuous Galerkin or finite difference method.

We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex -gon, our construction produces 2 basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, by transforming and combining a set of ( + 1)/2 basis functions known to obtain quadratic convergence. The technique broadens the scope of the so-called 'serendipity' elements, previously studied only for quadrilateral and regular hexahedral meshes, by employing the theory of generalized barycentric coordinates. Uniform error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges. Numerical evidence is provided on a trapezoidal quadrilateral mesh, previously not amenable to serendipity constructions, and applications to adaptive meshing are discussed.

Let = () be a finite, connected graph with weighted edges. We are interested in the problem of finding a subset ⊂ of vertices and weights such that for functions that are 'smooth' with respect to the geometry of the graph; here ~ indicates that we want the right-hand side to be as close to the left-hand side as possible. The main application are problems where is known to vary smoothly over the underlying graph but is expensive to evaluate on even a single vertex. We prove an inequality showing that the integration problem can be rewritten as a geometric problem ('the optimal packing of heat balls'). We discuss how one would construct approximate solutions of the heat ball packing problem; numerical examples demonstrate the efficiency of the method.