Discontinuous Galerkin method for advection diffusion with nonsmooth velocity

We study the advection diffusion equation with nonsmooth velocity field. Often in advection dominated regimes in fluids or PDE-constrained optimization problems, the velocity field is in a Sobolev space weaker than Lipschitz functions, which poses challenges for numerics and analysis. To numerically solve the problem, we study a Hybridizable Discontinuous Galerkin Method (HDG) mixed with classical upwinding. We use analytical tools of renormalized solutions for transport developed by Boyer (2005) to prove that the discrete solution converges strongly to a renormalized solution of the transport equation as the mesh size and the diffusion coefficient go to 0, even in the presence of Dirichlet boundary conditions and boundary layers. This work is joint with Noel Walkington (CMU).