Monotone Numerical Schemes for a Dirichlet Problem for Elliptic Operators in Divergence Form

Abstract

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 $A(\msx) u(\msx) \: = \: μ(\msx), \quad \msx \in D$, 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 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.