A superconvergence analysis is presened for the Shortleyâ€“Weller finite difference approximation
of second-order linear elliptic equations with bounued or unbounded derivatives on a polygonal
domain. In this analysis, we first formulate the method as a special finite element/volume method.
We then analyze the convergence of the method in a finite element framework. An $O(h^{1. 5})$-order
or $O(h^2)$ superconvergence of the solution derivatives in a discrete $H^1$ norm is obtained. Numerical
experiments are provided to support the theoretical convergence rate obtained.