  Consider the eigenvalue problem of a linear second order elliptic operator:
  \begin{equation}
  \nonumber
  -D\Delta \varphi -2\alpha\nabla m(x)\cdot
\nabla\varphi+V(x)\varphi=\lambda\varphi\ \ \hbox{ in }\Omega, \end{equation}
complemented by the Dirichlet boundary condition or the following general Robin
boundary condition:
  $$
  \frac{\partial\varphi}{\partial n}+\beta(x)\varphi=0 \ \ \hbox{ on
}\partial\Omega,
  $$ where $\Omega\subset\mathbb{R}^N (N\geq1)$ is a bounded smooth domain,
$n(x)$ is the unit exterior normal to $\partial\Omega$ at $x\in\partial\Omega$,
$D>0$ and $\alpha>0$ are, respectively, the diffusion and advection
coefficients, $m\in C^2(\overline\Omega),\,V\in C(\overline\Omega)$, $\beta\in
C(\partial\Omega)$ are given functions, and $\beta$ allows to be positive,
sign-changing or negative.
  In \cite{PZZ2019}, the asymptotic behavior of the principal eigenvalue of the
above eigenvalue problem as $D\to0$ or $D\to\infty$ was studied. In this paper,
when $N\geq2$, under proper conditions on the advection function $m$, we
establish the asymptotic behavior of the principal eigenvalue as
$\alpha\to\infty$, and when $N=1$, we obtain a complete characterization for
such asymptotic behavior provided $m'$ changes sign at most finitely many
times. Our results complement or improve those in \cite{BHN2005,CL2008,PZ2018}
and also partially answer some questions raised in \cite{BHN2005}.
