Assume $b$ is a constant vector, $v \in V_h$ where $V_h$ is a space of functions and $\Omega$ is our domain of integration.
$$ \int_{\Omega} (b \cdot\nabla u)\,v=-\int_{\Omega} (b \cdot \nabla v )\,u\,+\,\int_{\partial \Omega} v\,(n\cdot bu) $$
How is Green's formula applied in the above integral?