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

Brittany Froese Hamfeldt; Tiago Salvador

doi:10.1007/s10915-017-0586-5

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

Brittany Froese Hamfeldt, Tiago Salvador

Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

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.

Original language	English (US)
Pages (from-to)	1282-1306
Number of pages	25
Journal	Journal of Scientific Computing
Volume	75
Issue number	3
DOIs	https://doi.org/10.1007/s10915-017-0586-5
State	Published - Jun 1 2018

All Science Journal Classification (ASJC) codes

Software
Theoretical Computer Science
Numerical Analysis
General Engineering
Computational Theory and Mathematics
Computational Mathematics
Applied Mathematics

Keywords

Adaptive methods
Finite difference methods
Fully nonlinear elliptic partial differential equations
Higher-order methods

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10.1007/s10915-017-0586-5

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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.",

keywords = "Adaptive methods, Finite difference methods, Fully nonlinear elliptic partial differential equations, Higher-order methods",

author = "Hamfeldt, {Brittany Froese} and Tiago Salvador",

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AU - Salvador, Tiago

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N2 - 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.

AB - 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.

KW - Adaptive methods

KW - Finite difference methods

KW - Fully nonlinear elliptic partial differential equations

KW - Higher-order methods

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JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

IS - 3

ER -

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

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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