Structure-preserving discretization of cross-diffusion systems

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Ilaria Perugia
University of Vienna
Thursday 5th December 2024
Aula Consiglio VII piano - Dipartimento di Matematica
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Several applications in physics, biology, and chemistry involve systems with multiple components, such as gas mixtures, competing population species, and reacting chemical substances. These problems are modeled with nonlinear reaction-diffusion equations that include cross-diffusion terms. Cross diffusion occurs when the flux of one component is driven by the gradient of another component. The main challenges in designing numerical methods for approximating nonlinear cross-diffusion systems are that the diffusion matrix may not be symmetric or positive semidefinite, and that a maximum principle may not be available. In this talk, we present numerical methods based on the boundedness-by-entropy framework introduced by A. Jüngel in 2015. Motivated by the inherent entropy structure of the PDE system, nonlinear transformations involving the entropy variable allow for the enforcement of positivity in the approximate solutions. Specifically, we focus on a Local Discontinuous Galerkin method. By appropriately introducing auxiliary variables, the problem is reformulated so that nonlinearities do not appear within differential operators or interface terms. This results in nonlinear operators that can be naturally evaluated in parallel. The method allows for arbitrary degrees of approximation in space, preserves boundedness of the physical unknowns without requiring postprocessing or slope limiters, and satisfies a discrete version of the entropy stability estimate of the continuous problem. Contatto:
Ilaria Perugia is professor of Numerics of Partial Differential Equations at the University of Vienna, Austria. She received her PhD in 1999 from the University of Milano, Italy, under the supervision of Franco Brezzi. She held posts at the University of Pavia (university researcher, associate professor, full professor), as well as visiting positions at the University of Minnesota and at ETH Zürich. She was appointed professor at the University of Vienna in 2013. Since 2016, she has been deputy director of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in Vienna. She is serving in the editorial boards of several journals in Numerical Analysis and Scientific Computing. Her research is concerned with finite element methods for the numerical approximation of partial differential equations. Her interests mainly focus on the design and the analysis of standard and non standard finite element methods (discontinuous Galerkin, virtual element methods, finite elements with operator-adapted basis functions, space-time methods).