Super-convergence in maximum norm 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 in maximum 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.
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Submitted on : Monday, August 28, 2017 - 11:35:30 PM
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Lisl Weynans. Super-convergence in maximum norm of the gradient for the Shortley-Weller method. [Research Report] RR-8757, INRIA Bordeaux; INRIA. 2017, pp.16. ⟨hal-01176994v3⟩

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