High-degree discontinuous finite element discrete quadrature sets for the Boltzmann transport equation

Abstract

This paper proposes a family of high-degree discontinuous finite element (DFEM) quadrature sets designed to improve accuracy of integrals over local angular domain in the discrete ordinates method. The angular domain is divided into three types of spherical meshes by inscribing the octahedron, icosahedron and cube, respectively. High-degree discontinuous finite element basis functions are generated over different spherical meshes to resolve associated quadrature weights, producing high-degree DFEM quadrature sets. The performance and accuracy of the DFEM quadrature family was tested in the integration of spherical harmonics functions and some transport problems. Results demonstrate that kth-degree DFEM quadrature sets with any quadrature order can exactly integrate up to all kth-degree spherical harmonics functions in the direction cosines over local angular domain. Results also suggest that the performance of high-degree DFEM quadrature sets is comparable or better than that of traditional quadrature sets for transport problems with unsmooth or nearly discontinuous angular flux solutions. Locally-refined DFEM quadrature sets can obtain the same accuracy with less expense compared with uniform quadrature sets. DFEM quadrature sets with different degrees can offer two types of local refinement by increasing the quadrature order and changing the degree of discontinuous finite element basis functions, which can be further expected for use in an angular adaptive algorithm.