Title: Discontinuous Galerkin Methods for Diffusion-Dominated Radiative Transfer Problems
Speaker: Prof. Dr. Guido Kanschat University of Heidelberg
Date: 11 March 2014
Time: 3:30 - 4:30 p.m.
Venue: Lecture Hall I, Department of Mathematics
While discontinuous Galerkin (DG) methods had been developed and analyzed in the 1970s and 80s with applications in radiative transfer and neutron transport in mind, it was pointed out later in the nuclear engineering community, that the upwind DG discretization by Reed and Hill may fail to produce physically relevant approximations, if the scattering mean free path length is smaller than the mesh size. Mathematical analysis reveals, that in this case, convergence is only achieved in a continuous subspace of the finite element space. Furthermore, if boundary conditions are not chosen isotropically, convergence can only be expected in relatively weak topology. While the latter result is a property of the transport model, asymptotic analysis reveals, that the forcing into a continuous subspace can be avoided. By choosing a weighted upwinding, the conditions on the diffusion limit can be weakened. It has been known for long time, that the so called diffusion limit of radiative transfer is the solution to a diffusion equation; it turns out, that by choosing the stabilization carefully, the DG method can yield either the LDG method or the method by Ern and Guermond in its diffusion limit. Finally, we will discuss an efficient and robust multigrid method for the resulting discrete problems.