Convergent Finite Difference Methods for Fully Nonlinear Elliptic Equations in Three Dimensions

Document Type

Article

Publication Date

1-1-2022

Abstract

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.

Identifier

85120774168 (Scopus)

Publication Title

Journal of Scientific Computing

External Full Text Location

https://doi.org/10.1007/s10915-021-01714-6

e-ISSN

15737691

ISSN

08857474

Issue

1

Volume

90

Grant

1751996

Fund Ref

Center for Hierarchical Manufacturing, National Science Foundation

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