Properties of Transport Solution for Neutron Density in Slab Geometry in Piecewise Polynomial Approximation

Abstract

The order of accuracy of finite-element type approximations, when applied to partial differential equations, depends generally on the smoothness of the exact solution, measured by means of Sobolev norms. The properties of the one-speed neutron density in multi-layer slab geometry are considered from this point of view. It is shown in particular that the neutron density admits finite Sobolev-integrals of order less than unity only. The integrability results are applied to derive asymptotic error estimates for piecewise polynomial variational approximations. The inclusion of proper singular functions in a polynomial trial space is shown to speed up the convergence.