Aula Magna, Dipartimento di Matematica
Many physical problems involving heterogeneous spatial scales, such as the flow-through fractured porous media, the study of fiber-reinforced materials, or the modeling of blood circulation in living tissues — just to mention a few examples — can be described as coupled partial differential equations defined in domains of heterogeneous dimensions that are embedded into each other. The definition and the approximation of coupling operators that are suitable for such problems remain challenging, both theoretically and computationally.
In this presentation, I will introduce a comprehensive mathematical framework for analyzing and approximating partial differential equations coupled with non-matching constraints across different scales, with a focus on using Lagrange multipliers for the enforcement. I will discuss the well-posedness, stability, and robustness of the problem with respect to the small scale, as well as its numerical approximation based on non-matching and immersed finite element methods, which provide a natural way to perform geometric dimensionality reduction.