Convergence Analysis in the Maximum Norm of the Numerical Gradient of the Shortley–Weller Method

The Shortley–Weller method is a standard central finite-difference-method for solving the Poisson equation in irregular domains with Dirichlet boundary conditions. It is well known that the Shortley–Weller method produces second-order accurate solutions and it has been numerically observed that the solution gradients are also second-order accurate; a property known as super-convergence. The super-convergence was proved in the L² norm in Yoon and Min (J Sci Comput 67(2):602–617, 2016). In this article, we present a proof for the super-convergence in the L^∞ norm.