A proof in the finite-difference spirit of the superconvergence of the gradient for the Shortley-Weller method

Lisl Weynans 1, 2
1 MEMPHIS - Modeling Enablers for Multi-PHysics and InteractionS
Inria Bordeaux - Sud-Ouest, IMB - Institut de Mathématiques de Bordeaux
Abstract : We prove in this paper the second-order super-convergence of the gradient for the Shortley-Weller method. Indeed, with this method the discrete gradient is known to converge with second-order accuracy even if the solution itself only converges with second-order. We present a proof in the finite-difference spirit, inspired by the paper of Ciarlet [2] and taking advantage of a discrete maximum principle to obtain estimates on the coefficients of the inverse matrix. This reasoning leads us to prove third-order convergence for the numerical solution near the boundary of the domain, and then second-order convergence for the discrete gradient in the whole domain. The advantage of this finite-difference approach is that it can provide locally pointwise convergence results depending on the local truncation error and the location on the computational domain, as well as convergence results in maximum norm.
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Submitted on : Thursday, July 16, 2015 - 1:24:02 PM
Last modification on : Thursday, March 21, 2019 - 11:04:06 AM
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Lisl Weynans. A proof in the finite-difference spirit of the superconvergence of the gradient for the Shortley-Weller method. [Research Report] RR-8757, INRIA Bordeaux; INRIA. 2015, pp.14. ⟨hal-01176994v1⟩

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