The convergence properties of the discontinuous Galerkin finite element method (DGFEM) applied to the transport equation are presented for variants of Larsen's test case. The analysis is performed for two-dimensional structured and unstructured triangular meshes, with DGFEM approximations up to order 4. Pure absorber media and scattering media are considered. The influence of the mesh alignment with the singularities of the transport solution is described. The numerically observed convergence rates are related to theoretical results.