This paper presents a superconvergence analysis for the Shortley-Weller finite difference approximation of second-order self-adjoint elliptic equations with unbounded derivatives on a polygonal domain with the mixed type of boundary conditions. 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 superconvergence of the solution derivatives in a discrete H-1 norm is obtained. Finally, numerical experiments art provided to support the theoretical convergence rate obtained.
Li, Z-C., Fang, Q., Wang, S., & Hu, H-Y. (2008). Superconvergence of Solution Derivatives of the Shortley-Weller Difference Approximation to Elliptic Equations with Singularities involving the Mixed Type of Boundary Conditions. Numerical Functional Analysis and Optimization, 29(1-2), 161-196. https://doi.org/10.1080/01630560701872490