I am looking for the right integration by parts with the Green-Gauss divergence theorem applied on an advective Galerkin FEM term. I already did the transformation, but I want to be sure, if it is either right or wrong? My guiding example is from Donea and Huerta in 'FEM for Flow Problems' (Wiley, 2003) on chapter 3.6.2.2 'spatial discreziation' on page 109. Please correct me or add the right formulation.
$ \int w (v \cdot \nabla)^2 u d \Omega = - \int (v \cdot \nabla w, v \cdot \nabla u) d \Omega + \int (( v \cdot n) w , v \cdot \nabla u) d \Gamma $
$w$ - weight function, $v$ - velocity vector, $u$ - field parameter, $n$ - normal vector, $\Omega$ - element area, $\Gamma$ - element boundary
Big thanks in advance!