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arXiv · 2506.02595

Numerical methods for fully nonlinear degenerate diffusions

Edgard A. Pimentel·Ercília Sousa
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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, a two-parameters regularisation entails new, genuine difficulties. To bypass them, we resort to the intrinsic properties of the regularised problem and a fixed-point type of argument. 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.

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Edgard A. Pimentel, Ercília Sousa. 2025-06-03. Numerical methods for fully nonlinear degenerate diffusions. https://arxiv.org/abs/2506.02595

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