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

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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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Acknowledgements

We thank Adam Oberman for helpful discussions and support of this project.

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Authors and Affiliations

  1. Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ, 07102, USA

    Brittany Froese Hamfeldt

  2. Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI, 48109, USA

    Tiago Salvador

Authors
  1. Brittany Froese Hamfeldt
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  2. Tiago Salvador
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Correspondence to Brittany Froese Hamfeldt.

Additional information

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).

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Hamfeldt, B.F., Salvador, T. Higher-Order Adaptive Finite Difference Methods for Fully Nonlinear Elliptic Equations. J Sci Comput 75, 1282–1306 (2018). https://doi.org/10.1007/s10915-017-0586-5

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  • DOI: https://doi.org/10.1007/s10915-017-0586-5

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