A nonconforming finite element approximation of the transport equation in quadrangular and hexagonal geometries

This paper introduces a new finite element approximation for multi-dimensional transport problems in piecewise homogeneous media. The transport equation is solved using a Galerkin technique with polynomial basis functions in space-angle variables derived from asymptotic transport theory. The phase space is partitioned into cells consistent with the geometry and having each an elemental expansion which is not a tensor product. Improved accuracy may be obtained by multiplying the number of cells or/and increasing the polynomial degree. Numerical results on 1D and 2D reference problems in square geometry show a good agreement with other approximate methods.