Sparse tensor spherical harmonics approximation in radiative transfer

The stationary monochromatic radiative transfer equation is a partial differential transport equation stated on a five-dimensional phase space. To obtain a well-posed problem, inflow boundary conditions have to be prescribed. The sparse tensor product discretization has been successfully applied to finite element methods in radiative transfer with wavelet discretization of the angular domain (Widmer 2009). In this report we show that the sparse tensor product discretization can be combined with a spectral discretization of the angular domain using spherical harmonics. Neglecting boundary conditions, we prove that the convergence rate of our method in terms of number of degrees of freedom is essentially the same as the convergence of the full tensor product method up to a logarithmic factor. For the case with boundary conditions, we propose a splitting of the physical function space and a conforming tensorization. Numerical experiments in two physical and one angular dimension show evidence for the theoretical convergence rates to hold in the latter case as well.