TY - JOUR
T1 - Convergent Finite Difference Methods for Fully Nonlinear Elliptic Equations in Three Dimensions
AU - Hamfeldt, Brittany Froese
AU - Lesniewski, Jacob
N1 - Funding Information:
The first author was partially supported by NSF DMS-1619807 and NSF DMS-1751996. The second author was partially supported by NSF DMS-1619807.
Funding Information:
This research was supported by NSF DMS-1619807 and NSF DMS-1751996
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2022/1
Y1 - 2022/1
N2 - We introduce a generalized finite difference method for solving a large range of fully nonlinear elliptic partial differential equations in three dimensions. Methods are based on Cartesian grids, augmented by additional points carefully placed along the boundary at high resolution. We introduce and analyze a least-squares approach to building consistent, monotone approximations of second directional derivatives on these grids. We then show how to efficiently approximate functions of the eigenvalues of the Hessian through a multi-level discretization of orthogonal coordinate frames in R3. The resulting schemes are monotone and fit within many recently developed convergence frameworks for fully nonlinear elliptic equations including non-classical Dirichlet problems that admit discontinuous solutions, Monge–Ampère type equations in optimal transport, and eigenvalue problems involving nonlinear elliptic operators. Computational examples demonstrate the success of this method on a wide range of challenging examples.
AB - We introduce a generalized finite difference method for solving a large range of fully nonlinear elliptic partial differential equations in three dimensions. Methods are based on Cartesian grids, augmented by additional points carefully placed along the boundary at high resolution. We introduce and analyze a least-squares approach to building consistent, monotone approximations of second directional derivatives on these grids. We then show how to efficiently approximate functions of the eigenvalues of the Hessian through a multi-level discretization of orthogonal coordinate frames in R3. The resulting schemes are monotone and fit within many recently developed convergence frameworks for fully nonlinear elliptic equations including non-classical Dirichlet problems that admit discontinuous solutions, Monge–Ampère type equations in optimal transport, and eigenvalue problems involving nonlinear elliptic operators. Computational examples demonstrate the success of this method on a wide range of challenging examples.
KW - Fully nonlinear elliptic equations
KW - Generalized finite difference methods
KW - Three dimensions
KW - Viscosity solutions
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U2 - 10.1007/s10915-021-01714-6
DO - 10.1007/s10915-021-01714-6
M3 - Article
AN - SCOPUS:85120774168
SN - 0885-7474
VL - 90
JO - Journal of Scientific Computing
JF - Journal of Scientific Computing
IS - 1
M1 - 35
ER -