Meshfree finite difference approximations for functions of the eigenvalues of the Hessian
This work provides a novel numerical framework for solving nonlinear elliptic PDEs on complicated domains with non-uniform point distributions, addressing a key bottleneck in mesh-based methods.
The paper introduces meshfree finite difference methods for approximating nonlinear elliptic operators depending on second directional derivatives or Hessian eigenvalues, achieving monotone schemes that converge to viscosity solutions on unstructured point clouds. Numerical experiments show convergence for complex domains, degenerate equations, and singular solutions.
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.