Meshfree finite difference approximations for functions of the eigenvalues of the Hessian

TL;DR

This paper presents meshfree finite difference methods on unstructured point clouds for approximating nonlinear elliptic PDEs involving Hessian eigenvalues, ensuring convergence to viscosity solutions.

Contribution

Introduction of monotone meshfree finite difference schemes for complex domains and eigenvalue-dependent operators, with proven convergence guarantees.

Findings

Methods converge on complex and random point clouds

Applicable to degenerate and singular PDEs

Numerical experiments confirm theoretical convergence

Abstract

We introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the Hessian. Approximations are defined on unstructured point clouds, which allows for very complicated domains and a non-uniform distribution of discretisation points. The schemes are monotone, which ensures that they converge to the viscosity solution of the underlying PDE as long as the equation has a comparison principle. Numerical experiments demonstrate convergence for a variety of equations including problems posed on random point clouds, complex domains, degenerate equations, and singular solutions.