A Least-Squares Finite Element Method for the Linear Boltzmann Equation with Anisotropic Scattering

Least-squares methods have been applied to a wide range of differential equations and have been established to be competitive with other existing discretization strategies [P. B. Bochev and M. D. Gunzburger, SIAM Rev., 40 (1998), pp. 789–837]. In this article, we consider a leastsquares method for the linear Boltzmann equation with anisotropic scattering. A similar method has already been developed, and extensively examined, for the linear Boltzmann equation with isotropic scattering. The success of the least-squares method for isotropic scattering depends on scaling the linear Boltzmann equation so that minimization of the least-squares functional in a discrete space always yields accurate discrete solutions. A similar scaling of the linear Boltzmann equation is employed for anisotropic scattering. In the previous work for isotropic scattering, coercivity and continuity results were established for the scaled least-squares functional relative to a physically reasonable norm. In this paper, we extend the previous coercivity and continuity results so that they hold in this more general case of anisotropic scattering. Additionally, we extend the bounds for the discretization error for the thin regime and for the thick regime. For the thick regime, we establish optimal error estimates for the case of highly anisotropic scattering.