A convergent finite difference method for computing minimal Lagrangian graphs

TL;DR

This paper introduces a convergent finite difference method for solving the nonlinear eigenvalue problem associated with minimal Lagrangian graphs, with potential applications in physics and materials science.

Contribution

A novel two-step generalized finite difference method is developed and proven to converge for computing minimal Lagrangian graphs, extending to Monge-Ampere equations.

Findings

Method successfully converges in challenging test cases

Numerical experiments validate stability and accuracy

Framework adaptable to optimal transport problems

Abstract

We consider the numerical construction of minimal Lagrangian graphs, which is related to recent applications in materials science, molecular engineering, and theoretical physics. It is known that this problem can be formulated as an eigenvalue problem for a fully nonlinear elliptic partial differential equation. We introduce and implement a two-step generalized finite difference method, which we prove converges to the solution of the eigenvalue problem. Numerical experiments validate this approach in a range of challenging settings. We further discuss the generalization of this new framework to Monge-Ampere type equations arising in optimal transport. This approach holds great promise for applications where the data does not naturally satisfy the mass balance condition, and for the design of numerical methods with improved stability properties.