Numerical methods for fully nonlinear degenerate diffusions

TL;DR

This paper develops finite difference methods for solving fully nonlinear degenerate elliptic equations, including free boundary problems, and proves their convergence with supporting numerical experiments.

Contribution

It introduces a regularisation-based approach to handle degeneracy and extends the methods to a broader class of non-variational problems.

Findings

Proved convergence of the proposed schemes.

Validated methods with numerical experiments.

Extended approach to non-variational problems.

Abstract

We propose finite difference methods for degenerate fully nonlinear elliptic equations and prove the convergence of the schemes. Our focus is on the pure equation and a related free boundary problem of transmission type. The cornerstone of our argument is a regularisation procedure. It decouples the degeneracy term from the elliptic operator driving the diffusion process. In the free boundary setting, the absence of degenerate ellipticity entails new, genuine difficulties. To bypass them, we resort to the intrinsic properties of the regularised problem. We present numerical experiments supporting our theoretical results. Our methods are flexible, and our approach can be extended to a broader class of non-variational problems.