Computer-assisted proofs for some nonlinear diffusion problems

In the last three decades, powerful computer-assisted techniques have been developed in order to validate a posteriori numerical solutions of semilinear elliptic problems of the form \(\Delta u +f(u,\nabla u) = 0\). By studying a well chosen fixed point problem defined around the numerical solution,...

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Veröffentlicht in:arXiv.org 2021-01
1. Verfasser: Breden, Maxime
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Sprache:eng
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Zusammenfassung:In the last three decades, powerful computer-assisted techniques have been developed in order to validate a posteriori numerical solutions of semilinear elliptic problems of the form \(\Delta u +f(u,\nabla u) = 0\). By studying a well chosen fixed point problem defined around the numerical solution, these techniques make it possible to prove the existence of a solution in an explicit (and usually small) neighborhood the numerical solution. In this work, we develop a similar approach for a broader class of systems, including nonlinear diffusion terms of the form \(\Delta \Phi(u)\). In particular, this enables us to obtain new results about steady states of a cross-diffusion system from population dynamics: the (non-triangular) SKT model. We also revisit the idea of automatic differentiation in the context of computer-assisted proof, and propose an alternative approach based on differential-algebraic equations.
ISSN:2331-8422
DOI:10.48550/arxiv.2102.01501