University of Oxford

Modelling the behaviour of light in photonic crystal fibres requires

solving 2nd-order elliptic eigenvalue problems with discontinuous

coefficients. The eigenfunctions of these problems have limited

regularity. Therefore, the planewave expansion method would appear to

be an unusual choice of method for such problems. In this talk I

examine the convergence properties of the planewave expansion method as

well as demonstrate that smoothing the coefficients in the problem (to

get more regularity) introduces another error and this cancels any

benefit that smoothing may have.

I will discuss the investigatation of highly nonlinear solitary waves in heterogeneous one-dimensional granular crystals using numerical computations, asymptotics, and experiments. I will focus primarily on periodic arrangements of particles in experiments in which stiffer/heavier stainless stee are alternated with softer/lighter ones.

The governing model, which is reminiscent of the Fermi-Pasta-Ulam lattice, consists of a set of coupled ordinary differential equations that incorporate Hertzian interactions between adjacent particles. My collaborators and I find good agreement between experiments and numerics and gain additional insight by constructing an exact compaction solution to a nonlinear partial differential equation derived using long-wavelength asymptotics. This research encompasses previously-studied examples as special cases and provides key insights into the influence of heterogeneous, periodic lattice on the properties of the solitary waves.

I will briefly discuss more recent work on lattices consisting of randomized arrangements of particles, optical versus acoustic modes, and the incorporation of dissipation.

This is not intended to be a systematic History, but a selection of highlights, with some digressions, including:

The early days of the Computing Lab;

How the coming of the Computer changed some of the ways we do Computation;

A problem from the Study Groups;

Influence of the computing environment (hardware and software);

Convergence analysis for the heat equation, then and now.

Travelling waves are highly symmetric solutions to the Hamiltonian lattice equation and are determined by nonlinear advance-delay differential equations. They provide much insight into the microscopic dynamics and are moreover fundamental building blocks for macroscopic

lattice theories.

In this talk we concentrate on travelling waves in convex FPU chains and study both periodic waves (wave trains) and homoclinic waves (solitons). We present a new existence proof which combines variational and dynamical concepts.

In particular, we improve the known results by showing that the profile functions are unimodal and even.

Finally, we study the complete localization of wave trains and address additional complications that arise for heteroclinic waves (fronts).(joint work with Jens D.M. Rademacher, CWI Amsterdam)

The birational classification of varieties is an interesting and ongoing problem in algebraic geometry. This talk aims to give an

overview of the progress made on this problem in the special case where the varieties considered are surfaces in projective space.

I will begin by talking briefly about the Lavrentiev phenomenon and its implications for computations. In short, if a minimization problem exhibits a Lavrentiev gap then `naive' numerical methods cannot be used to solve it. In the past, several regularization techniques have been used to overcome this difficulty. I will briefly mention them and discuss their strengths and weaknesses.

The main part of the talk will be concerned with a class of convex problems, and I will show that for this class, relatively simple numerical methods, namely (i) the Crouzeix--Raviart FEM and (ii) the P2-FEM with under-integration, can successfully overcome the Lavrentiev gap.