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24-10-2017 | Issue 3/2018

Journal of Scientific Computing 3/2018

Higher-Order Adaptive Finite Difference Methods for Fully Nonlinear Elliptic Equations

Journal:
Journal of Scientific Computing > Issue 3/2018
Authors:
Brittany Froese Hamfeldt, Tiago Salvador
Important notes
The first author was partially supported by NSF DMS-1619807. The second author was partially supported by NSERC Discovery Grant RGPIN-2016-03922 and by Fundação para a Ciência e Tecnologia (FCT) Doctoral Grant (SFRH/BD/84041/2012).

Abstract

We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for adaptive meshes and complicated geometries, while still ensuring consistency, monotonicity, and convergence. We describe an algorithm for efficiently computing the non-traditional finite difference stencils. We also present a strategy for computing formally higher-order convergent methods. Computational examples demonstrate the efficiency, accuracy, and flexibility of the methods.

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