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arXiv · 2509.14150

A numerical scheme for a fully nonlinear free boundary problem

Edgard A. Pimentel·Ercília Sousa
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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.

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Edgard A. Pimentel, Ercília Sousa. 2025-09-17. A numerical scheme for a fully nonlinear free boundary problem. https://arxiv.org/abs/2509.14150

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