Implementation details

For a cell with volume $V$, Gauss' theorem is applied to give:

$$\div \left( \Gamma \grad \phi \right) = \frac{1}{V}\int_V \div \left( \Gamma \grad \phi \right)dV = \frac{1}{V} \oint_S \Gamma \grad \phi d\vec{S} = \frac{1}{V} \sum_i^{\mathrm{nFace}}\left(\grad{\phi}\right)_{f,i} \dprod \Gamma_f \vec{S}_{f,i}$$

i.e. the laplacian is given by the sum over all faces of the dot product of the face normal with the gradient value at the face. To transform the quantity from the cell centre to the face centre, an interpolation scheme is required.

See