Convergence of meshfree collocation methods for fully nonlinear parabolic equations

TL;DR

This paper proves that meshfree collocation methods converge for solving fully nonlinear parabolic PDEs using viscosity solutions, supported by a numerical experiment with radial basis functions.

Contribution

It establishes the convergence of meshfree collocation methods for nonlinear parabolic PDEs within the viscosity solutions framework, a novel theoretical result.

Findings

Convergence of meshfree collocation methods is proven.

Numerical experiments confirm the theoretical convergence.

Radial basis functions effectively demonstrate the method's applicability.

Abstract

We prove the convergence of meshfree collocation methods for the terminal value problems of fully nonlinear parabolic partial differential equations in the framework of viscosity solutions, provided that the basis function approximations of the terminal condition and the nonlinearities are successful at each time step. A numerical experiment with a radial basis function demonstrates the convergence property.