Then, the chain rule and the \textsc{Green-Gaussian} integral theorem are applied.
For standard Galerkin schemes, the weighting functions $W_j$ are chosen the same as the ansatz functions $N_j$. However, this does not yield a locally mass-conservative scheme.
Therefore, for the Box method, the weighting functions $W_j$ are chosen as
where $V_i$ is the volume of the FV box $B_i$ associated with node $i$.
The application of this assumption in combination with
$\int_\Omega W_j \:q \: dx = V_i \: q$ yields
Thus, the Box method is a Petrov-Galerkin scheme, where the weighting functions do not belong to the same function space than the ansatz functions.
Inserting definition \eqref{eq:weightingFunctions} into equation \eqref{eq:weightedResidual} and using the \textsc{Green-Gaussian} integral theorem results in