Hypocoercivity-Preserving Discretisations

Numerous physical, chemical, biological and social dynamic processes are often characterised by convergence to long-time equilibria. In many important cases the diffusion/dissipation required to arrive to such equilibria is explicitly present in some of the spatial directions only. 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 directions in which no dissipation appears explicitly in the model. In his celebrated AMS memoir, Villani introduced the eponymous concept of `Hypocoercivity' to describe a framework able to explain decay to equilibrium in the presence of dissipation in some directions only.

The broad objective of this project is the development of hypocoercivity-preserving Galerkin discretisations for very general classes of diffusion-degenerate kinetic problems. The construction of hypocoercivity-preserving variational methods unlock the potential of porting the already rich methodology of Galerkin finite element methods for standard, non-degenerate PDEs to large classes of hypocoercive problems.

To this end, we shall develop a general variational framework of non-conforming Galerkin methods that are able to counteract the inconsistency arising by differentiation of Galerkin spaces of reduced global regularity. This will be achieved by addressing the key challenge of lack of commutativity between differentiation and discretisation in the context of mesh-based Galerkin-type numerical methods via the use of carefully constructed non-conforming weak formulations of the underlying evolution problems. This will enable the development of discrete versions of hypocoercivity ensuring the accuracy and stability of the discretisations and provide convergence rates. Further, appropriate reconstructions of these, typically non-conforming, Galerkin approximations, will enable the proof of the first rigorous a posteriori error bounds. The latter will, in turn, allow for mathematically justifiable adaptive algorithms to be developed, aiming to reduce the significant computational complexity of the numerical methods, due to their inherent high-dimensionality.

Such numerical methods will be suitable for arbitrarily long-time simulations of complex phenomena modelled by kinetic-type formulations. As a result, we will be able to offer unprecedented numerical capabilities. This will, in turn, lead to a new class of simulators for important physical and industrial processes ranging from plasma physics, to rarefied gas dynamics and to nuclear reactor safety simulations.