On strong convergence of an elliptic regularization with the Neumann boundary condition applied to a stationary advection equation

We consider a boundary value problem of a stationary advection equation with the homogeneous inflow boundary condition in a bounded domain with Lipschitz boundary, and consider its perturbation by $\epsilon \Delta$, where $\epsilon$ is a positive parameter and $\Delta$ is the Laplacian. In this article, we show the $L^2$ strong convergence of solutions as the parameter $\epsilon$ tends to $0$, and discuss its convergence rates assuming $H^1$ or $H^2$ regularity for original solutions. A key observation is that the convergence rate depends on the regularity of original solutions and a relation between the boundary and the advection vector field. Some numerical computations support optimality of our convergence estimates.