Postgraduate Numerical Analysis Day 2013


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 bulk-interface coupled system, with a moving interface governed by a fourth-order 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 re-meshing 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): 4281-4307.
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): 738-772.
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): 6270-6299.
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 mesh-size h and the wavenumber k must be respected in order to achieve discrete well-posedness 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, well-posedness 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) - DG-FEM approximation for Hamilton-Jacobi-Bellman equations with Cordes coefficients
Hamilton--Jacobi--Bellman (HJB) equations are fully nonlinear second order PDE that arise in the study of optimal control of stochastic processes. It is a long-standing problem to construct numerical methods that are consistent, stable and high-order for these equations, since existing monotone methods are necessarily low-order. We present a novel hp-version 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 4D-Variational Data Assimilation
4D-Variational 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) - Semi-Parametric 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 atmosphere-ocean 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 (low-dimensional) 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 semi-parametric 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
on the time dependent domain (a(t),b(t)). We outline a velocity-based 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 least-squares 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.