(joint work with H. Heumann, K. Li, C. Pagliantini, J. Xu) We consider boundary value problems for the advection-diffusion of differential forms on a bounded domain. Taking the cue from discretization of scalar advection-diffusion problems, we extend and investigate two discretizations. (i) Semi-Lagrangian approach: We discretize the material derivative by means of backward finite differences along the flow lines and obtain asymptotic L2-estimates for the resulting scheme. (ii) Stabilized Galerkin approach: We pursue an Eulerian discretization in the spirit of discontinuous Galerkin (DG) methods with upwind numerical flux. Proofs of convergence are provided for the limit case of pure advection. The scheme can even be extended to discontinuous velocities and performs well in numerical experiments.