P_N-FEM method

This chapter describes various stabilized finite element methods for solving the first-order transport equation with spherical harmonics angular approximation. All of these methods are unified to the Petrov-Galerkin framework. The weak form of the stabilized variational formulation is modified by adding a stabilization term which can be written as the inner product of a modified test functions and a scaled residual of the source within each element. The scaling (stabilization) parameters have significant impact on the stability and convergence of the equation, which can be identified by the variational multiscale (VMS) finite element method and discrete maximum principles (DMP). In this study, the scaling parameters are identified as intrinsic free path (IFP) scale, which are related to the removal cross sections and the characteristic length of the element. The discrete process of spatial and angular and the implementation of the finite element method are discussed in detail. Various benchmark problems have been calculated, and the numerical results show that these methods have high accuracy, stability, and void applicability.