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

TL;DR

This paper presents a new finite difference method for solving complex fully nonlinear elliptic PDEs in three dimensions, combining boundary point refinement and eigenvalue approximation to ensure convergence and accuracy.

Contribution

The authors develop a generalized, monotone finite difference scheme for 3D nonlinear elliptic equations, incorporating boundary refinement and eigenvalue discretization, advancing numerical solution techniques.

Findings

Method successfully solves diverse nonlinear elliptic PDEs in 3D.

Scheme guarantees convergence within existing theoretical frameworks.

Computational tests demonstrate robustness and accuracy on challenging problems.

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 $\mathbb{R}^3$. 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-Amp\`ere 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.