A Moment-Parity Multigrid Preconditioner for the First-Order System Least-Squares Formulation of the Boltzmann Transport Equation

This paper describes a preconditioned conjugate gradient scheme for the Pn-h finite element discretization of a first-order system least-squares (FOSLS) formulation of the Boltzmann transport equation. The preconditioner is based on the norm equivalence between the FOSLS functional and a V norm. Its realization is an inexact inversion of the system of partial differential equations corresponding to this V norm. This preconditioner is essentially a moment-parity multigrid solver. It involves a 2 × 2 block diagonal system with each block describing only the interior like-parity coupling (even-even or odd-odd) of the spherical harmonic coefficients or moments. The interior cross-parity coupling is handled in the outer conjugate gradient iteration. Since the like-parity coupling consists of only second- and zeroth-order differential terms, whereas the full Pn system consists also of cross-parity, first-order coupling terms, the construction of a robust multigrid algorithm for each diagonal block is easier than the construction for the full Pn system. Numerical results indicate that this preconditioned conjugate gradient algorithm is more robust than a stand-alone multigrid solver for the full Pn system.