arXiv:2611.14329v1 Announce Type: new Abstract: The convection-diffusion problem has been studied extensively as an important model for convection-dominated problems. The problem becomes more challenging when the convecting velocity is no longer Lipschitz and merely has Sobolev regularity. In this paper, we use Boyer's extension of the DiPerna--Lions theory for the transport equation to study the vanishing diffusion limit of discretizations of the convection-diffusion equation with Sobolev velocity. In the case of flux-adapted boundary conditions, we prove that a continuous Galerkin discretization converges strongly to the renormalized solution of the transport equation as the diffusion coefficient and mesh size go to zero, without requiring any relationship between them. We refer to this unconditional strong convergence as asymptotic robustness. In the case of Dirichlet boundary conditions, we use the renormalization theory and barrier arguments to prove that a piecewise constant hybridizable discontinuous Galerkin scheme is asymptotically robust, while the continuous Galerkin scheme may fail to converge. Our results indicate that the asymptotic robustness of schemes for the convection-diffusion problem depends on the realization of the boundary conditions as opposed to internal layers.