Monotone numerical schemes for a Dirichlet problem for elliptic operators in divergence form

We consider a second‐order differential operator A(x)=−∑ di,j=1∂iaij(x)∂j+ ∑ dj=1∂j(bj(x)·)+c(x) on ℝd, on a bounded domain D with Dirichlet boundary conditions on ∂D, under mild assumptions on the coefficients of the diffusion tensor aij. The object is to construct monotone numerical schemes to approximate the solution of the problem A(x)u(x)=µ(x), x∈D, where µ 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 possessing 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 21‐convergence in detail, and mention convergence in C and L1 spaces. We conclude by a numerical example illustrating the schemes and convergence results. Copyright © 2008 John Wiley & Sons, Ltd.