TY - JOUR
T1 - Higher-Order Adaptive Finite Difference Methods for Fully Nonlinear Elliptic Equations
AU - Hamfeldt, Brittany Froese
AU - Salvador, Tiago
N1 - Funding Information:
We thank Adam Oberman for helpful discussions and support of this project. 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).
Publisher Copyright:
© 2017, Springer Science+Business Media, LLC.
PY - 2018/6/1
Y1 - 2018/6/1
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
UR - http://www.scopus.com/inward/record.url?scp=85032029724&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85032029724&partnerID=8YFLogxK
U2 - 10.1007/s10915-017-0586-5
DO - 10.1007/s10915-017-0586-5
M3 - Article
AN - SCOPUS:85032029724
SN - 0885-7474
VL - 75
SP - 1282
EP - 1306
JO - Journal of Scientific Computing
JF - Journal of Scientific Computing
IS - 3
ER -