# Super-Convergence in Maximum Norm of the Gradient for the Shortley–Weller Method

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## Abstract

We prove in this paper the second-order super-convergence in \(L^{\infty }\)-norm of the gradient for the Shortley–Weller method. Indeed, this method is known to be second-order accurate for the solution itself and for the discrete gradient, although its consistency error near the boundary is only first-order. We present a proof in the finite-difference spirit, using a discrete maximum principle to obtain estimates on the coefficients of the inverse matrix. The proof is based on a discrete Poisson equation for the discrete gradient, with second-order accurate Dirichlet boundary conditions. The advantage of this finite-difference approach is that it can provide pointwise convergence results depending on the local consistency error and the location on the computational domain.

## Keywords

Finite-difference Poisson equation Super-convergence Discrete Green’s function Shortley–Weller method## Supplementary material

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