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Superconvergent Cartesian Methods for Poisson type Equations in 2D–domains

Olivier Gallinato 1, 2 Clair Poignard 2, 1
1 MONC - Modélisation Mathématique pour l'Oncologie
IMB - Institut de Mathématiques de Bordeaux, Institut Bergonié [Bordeaux], Inria Bordeaux - Sud-Ouest
Abstract : In this paper, we present three superconvergent Finite Difference methods on Cartesian grids for Poisson type equations with Dirichlet, Neumann or Robin conditions. Our methods are based on finite differences and high-order discretizations of the Laplace operator, to reach the superconvergence properties, in the sense that the first-order (and possibly the second-order) derivatives of the numerical solution are computed at the same order as the solution itself. We exhibit the numerical conditions that have to be fulfilled by the schemes to get such superconvergences and extensively illustrate our purpose by numerical simulations. We conclude by applying our method to a free boundary problem for cell protrusion formation recently proposed by the authors and colleagues. Note that quasistatic Stefan-like problem can be accurately solved by our methods.
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Submitted on : Thursday, November 12, 2015 - 3:22:55 PM
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Olivier Gallinato, Clair Poignard. Superconvergent Cartesian Methods for Poisson type Equations in 2D–domains. [Research Report] RR-8809, INRIA; Institut de Mathématiques de Bordeaux; Université de Bordeaux. 2015, pp.33. ⟨hal-01228046⟩

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