Oxford

Not much is known about the Chow rings  of moduli spaces of abelian varieties or more general Shimura varieties. The tautological ring of a Shimura variety of Hodge type is a subring of its Chow ring containing many "interesting" classes. I will talk about joint work with Torsten Wedhorn on this ring as well as its characteristic p variant. The later is strongly related to the question of understanding the cycle classes of Ekedahl-Oort strata in the Chow ring.

Wrinkling is a universal instability occurring in a wide variety of engineering and biological materials. It has been studied extensively for many different systems but a full description is still lacking. Here, we provide a systematic analysis of the wrinkling of a thin hyperelastic film over a substrate in plane strain using stream functions. For comparison, we assume that wrinkling is generated either by the isotropic growth of the film or by the lateral compression of the entire system. We perform an exhaustive linear analysis of the wrinkling problem for all stiffness ratios and under a variety of additional boundary and material effects.

When approximating PDEs with the finite element method, large sparse linear systems must be solved. The ideal preconditioner 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.

Achieving this for the stationary Navier-Stokes has proven challenging: LU factorisation is Reynolds-robust but scales poorly with degree of freedom count, while Schur complement approximations such as PCD and LSC degrade as the Reynolds number is increased.

Building on the work of Schöberl, Olshanskii and Benzi, in this talk we present the first preconditioner for the Newton linearisation of the stationary Navier--Stokes equations in three dimensions that achieves both optimal complexity and Reynolds-robustness. The scheme combines a novel tailored finite element discretisation, discrete augmented Lagrangian stabilisation, a custom prolongation operator involving local solves on coarse cells, and an additive patchwise relaxation on each
level. We present 3D simulations with over one billion degrees of freedom with robust performance from Reynolds number 10 to 5000.

Employing the usual multilevel Monte Carlo estimator, we introduce a framework for estimating the solutions of SDEs by an Euler-Maruyama scheme. By considering the expected value of such solutions, we produce simulations using approximately normal random variables, and recover the estimate from the exact normal distribution by use of a multilevel correction, leading to faster simulations without loss of accuracy. We will also highlight this concept in the framework of reduced precision and vectorised computations.

In this talk I will introduce a new mathematical model for the computational modelling of the active contraction of cardiac tissue using stress-assisted conductivity as the main mechanism for mechanoelectrical feedback. The coupling variable is the Kirchhoff stress and so the equations of hyperelasticity are written in mixed form and a suitable finite element formulation is proposed. Next I will introduce a simplified version of the coupled system, focusing on its analysis in terms of solvability and stability of continuous and discrete mixed-primal formulations, and the convergence of these methods will be illustrated through two numerical tests.

In petroleum reservoir simulation, the standard preconditioner is a two-stage process which involves solving a restricted pressure system with AMG. Initially designed for isothermal models, this approach is often used in the thermal case. However, it does not incorporate heat diffusion or the effects of temperature changes on fluid flow through viscosity and density. We seek to develop preconditioners which consider this cross-coupling between pressure and temperature. In order to study the effects of both pressure and temperature on fluid and heat flow, we first consider a model of non-isothermal single phase flow through porous media. For this model, we develop a block preconditioner with an efficient Schur complement approximation. Then, we extend this method for multiphase flow as a two-stage preconditioner.