Bienvenue - Laboratoire Jacques-Louis Lions

12 mars 2021 — 14h00
Exposé à distance retransmis par Zoom
Doug Arnold (Université du Minnesota, Minneapolis)
Complexes from complexes
Résumé
The finite element exterior calculus has highlighted the importance of Hilbert complexes to partial differential equations and their numerical solution. Hilbert complexes arise throughout mathematical physics. The fundamental partial differential operators from which most models in continuum physics are built may be realized as unbounded operators mapping between Sobolev and related Hilbert spaces, and these spaces and operators assemble into chain complexes. The resulting structure is a Hilbert complex : a finite sequence of Hilbert spaces together with closed unbounded operators from one space to the next such that the composition of two consecutive operators vanishes. This is a rich structure which combines functional analysis with homological algebra.
The most canonical and most extensively studied example of a Hilbert complex is the de Rham complex, which is what is required for application to fluid mechanics, electromagnetics, the Hodge Laplacian, and numerous other problems. This may give the impression that the finite element exterior calculus is only another way to look at these applications, which can be approached with less complicated machinery. But, in fact, there are many other important differential complexes as well, with applications to elasticity, plates, incompressible flow, general relativity, and other areas. These complexes are less well known and in many cases their properties, including properties needed to fit them into the finite element exterior calculus framework, have not been established.
In this talk I will discuss a systematic procedure for deriving such complexes and establishing their crucial properties.