1-Feb-2024, 12:00 (Dublin time)

Joint MACSI / INAF seminar, hybrid mode (University of Limerick, room A2002 / Zoom)

by Emmanuil (Manolis) Georgoulis, Heriot-Watt University, Edinburgh, UK / National Technical University of Athens, Greece

 “Hypocoercivity-preserving Galerkin discretisations of kinetic equations"

Numerous physical, chemical, biological, and social dynamic processes are characterised by convergence to long-time equilibria. These are often described as PDEs of kinetic type, whereby ``position'' and ``velocity'' are independent variables; well known examples of such are Kolmogorov and Fokker-Planck equations. These may also arise when modelling multi-agent interacting processes of particles, individuals, etc. In many important cases the diffusion/dissipation required to arrive to such equilibria is explicitly present in some of the spatial directions only, that is there exist evolution PDEs with degenerate diffusion yet converging to equilibrium states as time goes to infinity. This, somewhat counter-intuitive at first, state of affairs suggests that decay to equilibrium is due to finer hidden structure, which allows for the transport terms to also "propagate dissipation" to the spatial directions in which no dissipation appears explicitly in the PDE model. This property has been studied extensively by Villani who coined the term “hypocoercivity" to describe it in his celebrated 2009 AMS Memoir. 

In the talk I will present recent results on how to design and analyse hypocoercivity-preserving Galerkin discretisations, in an effort to port the concept of hypocoercivity in the design and analysis of cutting-edge numerical methods. To that end I plan to start by presenting the first provably hypocoercivity-preserving Galerkin (non-conforming) finite element method for the model problem of Kolmogorov equation and discuss some very recent results on a different approach for the same equation. I plan to conclude with some first results of Galerkin methods for the inhomogeneous  Fokker-Plack equation on exponentially weighted function spaces. Some of the results are joint work with Zhaonan Dong (INRIA, Paris) and Philip Herbert (Sussex).