Abstract: We consider finite element approximations of the stationary incompressible Navier-Stokes equations. An ideal preconditioner for the linear systems arising from these equations yields convergence that is algorithmically optimal and parameter robust, i.e. the number of Krylov iterations required to solve the linear system to a given accuracy does not grow substantially as the mesh or problem parameters are changed.
It has proven challenging to develop solvers that exhibit both properties; matrix factorisations are robust to Reynolds number but scale badly with dof count, whereas Schur complement based algorithms such as PCD and LSC scale linearly in the dof count but their performance decreases as the Reynolds number is increased
Building on the ideas of SchÃ¶berl, Benzi, and Olshanskii, we present an augmented Lagrangian based preconditioner with linear complexity and iteration counts that only grow mildly with respect to the Reynolds number. The key ingredient is a tailored multigrid scheme for the exactly divergence-free Scott-Vogelius discretisation consisting of custom smoothing and prolongation operators.
This work has been done in collaboration with Patrick Farrell, Lawrence Mitchell, and Ridgway Scott.
Abstract: I'll talk about my work in mathematical visualization: making accurate, effective, and beautiful pictures, models, and experiences of mathematical concepts. I'll discuss what it is that makes a visualization compelling, and show many examples in the medium of 3D printing, as well as some work in virtual reality and spherical video. I'll also discuss my experiences in teaching a project-based class on 3D printing for mathematics students.
Abstract: I'll start with a quickie course on von Neumann algebras, for which a suitable reference might be Vaughan Jones' course notes, available at https://math.vanderbilt.edu/jonesvf/. Then I will state several equivalent forms of the Connes Embedding Problem, for which a good reference is arXiv:1003.2076.
Abstract: Each year recently I have given a talk about some important topic that is not one of my research areas. This lecture concerns ideas in the science of nutrition and metabolism that I feel most people should know about. We seem to know more about planets circling other stars than about metabolism. And there are good reasons for that.