A superconvergence analysis is presened for the Shortley–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.