Bath/Brunel/Cardiff/Imperial/Oxford/Reading/UCL/Warwick
Postgraduate Numerical Analysis Day 2013
Abstracts
Postgraduate Numerical Analysis Day 2013, Bath
 Tom Ranner (Warwick)  Unfitted finite element methods
for surface partial differential equations

Surface partial differential equations have grown in popularity within the
last twenty years with applications in fluid mechanics, biology and material
sciences becoming increasingly common. A traditional approach is to
triangulate the surface and apply a finite element method based on the
piecewise linear surface. These methods rely on being able to generate an
accurate, regular triangulation which in practice may be difficult to find.
In this talk, we propose an alternative approach using a bulk finite element
space to solve an embedded version of the surface equations. We apply this
methodology to derive two methods for solving a Poisson equation and go on to
show how one may use similar methods to solve a heat equation on an evolving
surface.

Andrea Saconni (Imperial)  An Unfitted Finite Element Method for the
Approximation of Void Electromigration

Microelectronic circuits usually contain small voids or cracks, and if those
defects are large enough to sever the line, they cause an open circuit. We
present a numerical method for investigating the migration of voids in the
presence of both surface diffusion and electric loading. Our mathematical
model involves a bulkinterface coupled system, with a moving interface
governed by a fourthorder geometric evolution equation and a bulk where the
electric potential is computed. Thanks to a proper approximation of the
interface, equidistribution of its vertices is guaranteed, therefore no
remeshing is necessary for the interface. Numerical challenges include the
coupling of the two sets of equations and the proper definition of the bulk
mesh at each time step: the algorithm used to identify cut, inside and outside
elements, as well as local adaptivity are explained in detail. Various
examples are performed with a C++based code to demonstrate the accuracy of
the method.
J. W. Barrett, H. Garcke, and R. Nürnberg, "On the parametric
finite element approximation of evolving hypersurfaces in R3."
Journal of Computational Physics 227.9 (2008):
42814307.
J. W. Barrett, R. Nürnberg, and V. Styles, "Finite
element approximation of a phase field model for void
electromigration." SIAM journal on numerical analysis
42.2 (2004): 738772.
J. W. Barrett, H. Garcke, and R.
Nürnberg, "On stable parametric finite element methods for the
Stefan problem and the Mullins.Sekerka problem with applications to
dendritic growth." Journal of Computational Physics
229.18 (2010): 62706299.
 Luke Swift (UCL)  Stabilised finite element methods for
Helmholtz equation

In this talk we will discuss stabilised finite element methods for the
numerical solution of Helmholtz equation. It is well known that if the
standard Galerkin finite element method is used, certain conditions on the
meshsize h and the wavenumber k must be respected in order
to achieve discrete wellposedness and optimal error estimates. In particular,
for low order elements, the condition "k²h small" must be
respected. We will show that for the stabilised finite element methods,
wellposedness of the discrete system is guaranteed without condition on
h and k. In one space dimension for piecewise affine finite
elements it has also been shown that for suitably chosen stabilisation
parameters the pollution error in the error estimates can be eliminated. We
will review these results and then present some numerical evidence that
stabilisation may improve the solution quality also in two space dimensions.
 Iain Smears (Oxford)  DGFEM approximation for
HamiltonJacobiBellman equations with Cordes coefficients

HamiltonJacobiBellman (HJB) equations are fully nonlinear second order PDE
that arise in the study of optimal control of stochastic processes. It is a
longstanding problem to construct numerical methods that are consistent,
stable and highorder for these equations, since existing monotone methods are
necessarily loworder. We present a novel hpversion discontinuous Galerkin
FEM for HJB equations with coefficients that satisfy a Cordes condition. The
method is shown to be consistent and stable, with convergence rates that are
optimal with respect to mesh size, and suboptimal in the polynomial degree by
only half an order. Numerical experiments illustrate the accuracy and
computational efficiency of the scheme.
 Siân Jenkins (Bath)  Numerical Model Error in
4DVariational Data Assimilation

4DVariational data assimilation is typically used for forecasting physical
systems. It finds an initial condition for a numerical model, by combining
observations with predictions. The numerical model is then used to produce a
forecast. Numerical model error affects the accuracy of the initial condition
and its forecast. We find an upper bound for the numerical model error
introduced by finite difference schemes used to solve the linear advection
equation and analyse its order of convergence.

Sebastian Krumscheid (Imperial)  SemiParametric Estimation for Multiscale Diffusions

Most dynamical systems in the natural sciences are characterized by the
presence of processes that occur across several length and time scales.
Examples include the atmosphereocean system, biological systems, materials
and molecular dynamics. Typically only the dynamics at the macroscopic scale
is of main interest. While multiscale methods (averaging and homogenization)
provide the analytical framework for the rigorous derivation of effective
(lowdimensional) dynamics, statistical inference for these multiscale systems
remains far from being straightforward. In particular, standard statistical
techniques such as maximum likelihood become biased due to the multiscale
error. In this talk we will introduce a novel semiparametric estimation
procedure for multiscale diffusions that does not suffer from this bias. In
addition to presenting several illustrative examples that show the accuracy of
our method, we will present rigorous convergence results for a particular
simple model.
 Nicholas Bird (Reading)  High Order Nonlinear Diffusion

We examine the fourth order nonlinear diffusion equation
u_{t}=(u^{n}u_{xxx})_{x}
on the time dependent domain (a(t),b(t)). We outline a velocitybased
method of solution to this PDE in a moving framework. The method uses a local
conservation of mass principle and maintains scale invariance.
We next introduce a moving mesh Finite Difference method for use in obtaining
numerical solutions to the PDE. The method is designed such that when
n=1, using initial conditions sampled from a similarity solution and
through use of a scale invariant time stepping scheme the approximation
matches the exact solution to the problem to within rounding error for all
time.
The effect of choosing n≠1 on the method is highlighted and
possible extensions to the method are briefly discussed.
 Tom Croft (Cardiff)  Convergence of Proper Generalized
Decomposition Algorithms

We look at a new a family of methods called Proper Generalized
Decompositions (PGD) for the efficient approximation of solutions to PDEs
defined in high dimensional spaces. In particular we investigate the
convergence of such algorithms for the Poisson equation, for which a proof
already exists, and for the Stokes equations, which are slightly more
problematic. This is done by comparing the the PGD to a greedy algorithm, the
likes of which appear in nonlinear approximation theory. We firstly
introduce the simplest definition of the PGD which is a progressive algorithm
based on the Galerkin formulation of the associated problem and at the end we
shall look at PGD algorithms based on leastsquares formulations which provide
a more robust setting for proving convergence.
 Ross McKenzie (Cardiff)  Solving differential
algebraic equations by structural analysis and index
reduction

Differential algebraic equations, DAEs, appear frequently in applications
involving equation based modelling, from robotics to chemical engineering.
There is a concept of (differential) index which is a measure of how different
a DAE is from its corresponding ODE. An important way of making a DAE
amenable to numerical solution is by reducing the index to get a corresponding
ODE and using an ODE solution method. The signature matrix method developed by
Pryce does not rely on an index reduction step and instead solves the DAE
directly via Taylor series. The talk will draw comparisons between these two
different approaches and show the signature matrix method is in some sense
equivalent to the dummy derivative index reduction method developed by
Mattsson and S\"oderlind. There will be several examples used to illustrate
both methods and numerical results for the signature matrix method will be
presented.