In this talk, we discuss singular perturbation for linear first-order advection equations in a bounded domain, where a small diffusion term is added to the original problem. In particular, we consider the case of mixed boundary conditions, where the Dirichlet condition is imposed on the inflow boundary and the Neumann condition is imposed on the rest of the boundary.
Our focus is on the convergence of the perturbed solution to the original solution as the perturbation parameter tends to zero. We introduce some convergence estimates showing that the convergence rates depend on not only the regularity of the original solution but also the geometric relationship between the advection vector field and the boundary of the domain. We also show some numerical results that support the sharpness of our estimates.