I want to discretize the following equation using a Finite Volume Method $$\nabla \cdot (a(x)\nabla u)=f(x)\\x\in \Omega \subset \mathbb{R}^2 \\u_{|\partial\Omega}=g$$ I'm using Voronoi cells here: let $V_i \subset\Omega$, $i=1,\dots,N$, such that $\bigcup\limits_{i=1}^N V_i=\Omega$, $V_i\cap V_j=\partial V_i \cap \partial V_j$ for $i\ne j$ be the mesh cells.
After applying the Divergence theorem and integrating the equation over each $V_i$, obtain
$$\int_{V_i} \nabla \cdot (a(x)\nabla u) dx = \int_{\partial V_i} a(x)\nabla u \cdot n ds=\int_{V_i} fdx$$
Let $x_i$ be the "center" of $V_i$ (this is a cell-cenetered method). Define $n_i\subset \{1,\dots,N\}$ as the set of indices of the cells immediately neighbouring $V_i$. Let the boundary between any two mesh cells be defined as
$$\Gamma_{i,j}:=\partial V_i \cap \partial V_j$$
Define the length of the boundary segment between neighbouring cells $V_i$ and $V_j$ as $$l_{i,j}:=\left|\Gamma_{i,j}\right|, j\in n_i$$
and the distance between two cell "centers" as $$h_{i,j}:=\|x_i-x_j\|_2, j\in n_i$$
Approximate $a(x)$ for each two neighbouring cells $V_i,V_j$ as
$$a_{i,j}:=a\left(\frac{x_i+x_j}{2}\right), j\in n_i$$
So we obtain the following discretization:
$\int_{\partial V_i} a(x)\nabla u \cdot n ds=\sum_{j\in n_i}\int_{\Gamma_{i,j}}a(x)\nabla u(x)\cdot n ds \approx \sum_{j\in n_i} a_{i,j} \frac{u_j-u_i}{h_{i,j}}l_{i,j}=|V_i|f(x_i)\approx \int_{V_i} f(x)dx$
To discretize the Dirichlet BC using centered cells, as far as I understand, one needs to use outer ghost cells. So let $V_{G_j}$ be such a cell, adjacent to $V_j$ for some $j\in\{1,\dots, N\}$, and $x_{G_{j}}\in V_{G_{j}}$ be its "center". But this is where I'm stuck. Where do I go from here?
P.S. I know there is a similar question here, but I couldn't quite follow the notation there.