A Meshfree Method for Solving the Monge-Amp\`ere Equation

TL;DR

This paper introduces a meshfree collocation method for solving the 2D Monge-Ampère equation, demonstrating high convergence rates and providing both numerical and theoretical validation.

Contribution

It presents a novel meshfree collocation approach with polynomial trial spaces for fully nonlinear PDEs, specifically the Monge-Ampère equation.

Findings

Convergence rates can be up to exponential depending on solution smoothness

Numerical experiments confirm theoretical convergence rates

Method effectively solves the 2D Dirichlet problem for Monge-Ampère

Abstract

This paper solves the two-dimensional Dirichlet problem for the Monge-Amp\`ere equation by a strong meshless collocation technique that uses a polynomial trial space and collocation in the domain and on the boundary. Convergence rates may be up to exponential, depending on the smoothness of the true solution, and this is demonstrated numerically and proven theoretically, applying a sufficiently fine collocation discretization. A much more thorough investigation of meshless methods for fully nonlinear problems is in preparation.