Approximation theory for a certain class of operators and some of their applications to transport theory

The convergence of discrete-ordinates approximations under various assumptions on cross-section and source data has been shown in different function spaces by Anselone, Keller, Wendroff, Nestell, Madsen, Nelson, and Victory. For the establishment of the convergence theory of discrete-ordinates methods in higher spatial dimensions, we develop in this paper an approximation theory for a certain class of operators - - - naturally arising in conjunction with the transport equation. The theory developed in this paper is applied to proving that the critical parameter and critical flux computed by discrete-ordinate methods do converge to the corresponding quantities for the undiscretized, three-dimensional transport equation.