  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.
