A convergent finite difference method for computing minimal Lagrangian graphs

Abstract

<p style='text-indent:20px;'>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 additive 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-Ampère 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.</p>