  We consider a second order differential operator $A(\msx) = -\:\sum_{i,j=1}^d
\partial_i a_{ij}(\msx) \partial_j \:+\: \sum_{j=1}^d \partial_j \big(b_j(\msx)
\cdot \big)\:+\: c(\msx)$ on ${\bbR}^d$, on a bounded domain $D$ with Dirichlet
boundary conditions on $\partial D$, under mild assumptions on the coefficients
of the diffusion tensor $a_{ij}$. The object is to construct monotone numerical
schemes to approximate the solution to the problem $A(\msx) u(\msx) \: = \:
\mu(\msx), \quad \msx \in D$, where $\mu$ is a positive Radon measure. We start
by briefly mentioning questions of existence and uniqueness, introducing
function spaces needed to prove convergence results. Then, we define
non-standard stencils on grid-knots that lead to extended discretization
schemes by matrices possesing compartmental structure. We proceed to
discretization of elliptic operators, starting with constant diffusion tensor
and ending with operators in divergence form. Finally, we discuss
$W_2^1$-convergence in detail, and mention convergence in $C$ and $L_1$ spaces.
We conclude by a numerical example illustarting the schemes and convergence
results.
