# Applied Mathematics Seminars

Organisers: Ferran Brosa Planella and Thomasina Ball

The Applied Maths Seminars are held on Fridays 12:00-13.00. This year the seminar will be hybrid (at least for Term 1): you can choose to attend in person in room B3.02 or on MS Teams. The team for the seminar is the same as last year, but if you are not a member, you can send a membership request via MS Teams or email the organisers.

Please contact Ferran Brosa Planella or Thomasina Ball if you have any speaker suggestions for terms 2 and 3.

Seminar Etiquette: Here is a set of basic rules for the seminar.

• Please keep your microphone muted throughout the talk. If you want to ask a question, please raise your hand and the seminar organiser will (a) ask you to unmute if you are attending remotely or (b) get the speaker's attention and invite you to ask your question if you are in the room.
• If you are in the room with us, the room microphones capture anything you say very easily, and this is worth keeping in mind ☺️.
• You can choose to keep your camera on or not. Colleagues in the room will be able to see the online audience.
• Please let us know if you would like to meet and/or have lunch with any of the speakers who are coming to visit us so that I can make sure you have a place in the room.

### Term 1

#### Term 1

###### Week 2. Philip Herbert (Heriot-Watt) - Shape optimisation with Lipschitz functions

In this talk, we discuss a novel method in PDE constrained shape optimisation. We begin by introducing the concept of PDE constrained shape optimisation. While it is known that many shape optimisation problems have a solution, their approximation in a meaningful way is non-trivial. To find a minimiser, it is typical to use first order methods. The novel method we propose is to deform the shape with fields which are a direction of steepest descent in the topology of $W superscript {1, infinity}$. We present an analysis of this in a discrete setting along with the existence of directions of steepest descent. Several numerical experiments will be considered which compare a classical Hilbertian approach to this novel approach.

###### Week 3. Francis Aznaran (Oxford) - Finite element methods for the Stokes–Onsager–Stefan–Maxwell equations of multicomponent flow

The Onsager framework for linear irreversible thermodynamics provides a thermodynamically consistent model of mass transport in a phase consisting of multiple species, via the Stefan–Maxwell equations, but a complete description of the overall transport problem necessitates also solving the momentum equations for the flow velocity of the medium. We derive a novel nonlinear variational formulation of this coupling, called the (Navier–)Stokes–Onsager–Stefan–Maxwell system, which governs molecular diffusion and convection within a non-ideal, single-phase fluid composed of multiple species, in the regime of low Reynolds number in the steady state. We propose an appropriate Picard linearisation posed in a novel Sobolev space relating to the diffusional driving forces, and prove convergence of a structure-preserving finite element discretisation. This represents some of the first rigorous numerics for the coupling of multicomponent molecular diffusion with compressible convective flow. The broad applicability of our theory is illustrated with simulations of the centrifugal separation of noble gases and the microfluidic mixing of hydrocarbons. This is joint work with Alexander Van-Brunt.

###### Week 8. Eric Neiva (Collége de France & CNRS) - Unfitted finite element methods: decoupling the mesh from the geometry

The finite element method (FEM) approximates a PDE from a variational formulation of the problem. Its standard formulation requires a mesh fitting to the boundary of the geometry of interest. Yet, for many problems of practical interest, the geometry is so intricate that mesh generation requires frequent and time-consuming manual intervention. Boundary-fitted meshing can be avoided with unfitted or immersed FEMs. The main idea is to embed the geometry in a simple mesh (e.g., a Cartesian grid) and define the discretisation in the cells intersecting the geometry. In this talk, we will describe a novel unfitted FEM that circumvents the classical issue of immersed FEM: ill-conditioning due small cell-to-geometry intersections We will discuss its application to early embryo development in animals.

###### Week 10. TBC

Internet Access at Warwick:
Where possible, visitors should obtain an EDUROAM account from their own university to enable internet access whilst at Warwick.
If you need WiFi whilst at Warwick, click here for instructions (upon arrival at Warwick)
Registration:
You can register for any of the symposia or workshops online. To see which registrations are currently open and to submit a registration, please click here.
Contact:
Mathematics Research Centre
Zeeman Building
University of Warwick
Coventry CV4 7AL - UK
E-mail:
mrc@maths.warwick.ac.uk