A numerical scheme for a fully nonlinear free boundary problem
Abstract
We propose a numerical method to approximate viscosity solutions of fully nonlinear free transmission problems. The method discretises a two-layer regularisation of a PDE, involving a functional and a vanishing parameter. The former is handled via a fixed-point argument. We then prove that the numerical method converges to a one-parameter regularisation of the free boundary problem. Regularity estimates enable us to take the vanishing limit of such a parameter and recover a viscosity solution of the free transmission problem. Our main contribution is the design of a computational strategy, based on fixed-point arguments and approximated problems, to solve fully nonlinear free boundary models. We finish the paper with two numerical examples to validate our method.
Cite
@article{arxiv.2509.14150,
title = {A numerical scheme for a fully nonlinear free boundary problem},
author = {Edgard A. Pimentel and Ercília Sousa},
journal= {arXiv preprint arXiv:2509.14150},
year = {2025}
}