Global Error Bounds for the Petrov-Galerkin Discretization of the Neutron Transport Equation

In this paper, we prove that the numerical solution of the mono-directional neutron transport equation by the Petrov-Galerkin method converges to the true solution in the L2 norm at the rate of h2. Since consistency has been shown elsewhere, the focus here is on stability. We prove that the system of Petrov-Galerkin equations is stable by showing that the 2-norm of the inverse of the matrix for the system of equations is bounded by a number that is independent of the order of the matrix. This bound is equal to the length of the longest path that it takes a neutron to cross the domain in a straight line. A consequence of this bound is that the global error of the Petrov-Galerkin approximation is of the same order of h as the local truncation error. We use this result to explain the widely held observation that the solution of the Petrov-Galerkin method is second accurate for one class of problems, but is only first order accurate for another class of problems