## Applied Mathematics Seminars

### Programme

These seminars take place on Tuesdays, in Room M/2.06, Senghennydd Road, Cardiff from 3pm, unless otherwise stated.

When a seminar is not scheduled there is a collaborative workshop with other groups within the College of Physical Sciences & Engineering or a SIAM Chapter Meeting. Further details can be found on the School Diary.

For more information or if you wish to give a talk, please contact the programme organiser Dr Angela Mihai.

##### 7 October 2014

**Speaker:** Daniel Lesnic (University of Leeds).

**Title:** Determination of a force function in the wave equation.

**Abstract:** The determination of an unknown space- or time-dependent force function acting on a vibrating structure from boundary, interior or integral observations are investigated. Sufficient conditions for the uniqueness of solution are provided. These linear inverse force problems are ill-posed since small errors in the input data cause large errors in the output force solution. Consequently, when the input data is contaminated with noise, we use regularization methods, e.g. Tikhonov's regularization, or conjugate gradient method, in order to obtain a stable solution. Numerical results will be presented and discussed.

##### 14 October 2014

**Speaker:** Timo Betcke (University College London).

**Title:** A spectral view on convolution quadrature methods for the wave equation.

**Abstract:** Convolution Quadrature (CQ) methods are Laplace transform type methods for the solution of time-domain wave problems. They are particularly popular for the solution of exterior time-domain wave scattering problems using boundary integral equation formulations in the Laplace domain. In this talk we will take a spectral view on CQ methods and discuss the connection between scattering poles of the solution operator, the underlying time-stepping scheme and convergence of CQ methods. The presented numerical examples are computed with BEM++, an open-source boundary element library developed at UCL. We will give a brief overview of BEM++ and demonstrate its functionality for solving boundary integral equations.

##### 28 October 2014

**Speaker:** John Pryce (Cardiff School of Mathematics).

**Title:** The Forthcoming IEEE 1788 Standard for Interval Arithmetic.

**Abstract:** Interval arithmetic (IA) is the most used way of producing rigorously proven results in problems of continuous mathematics, usually in the form of real intervals that (even in presence of rounding error) are guaranteed to enclose a value of interest, such as a solution of a differential equation at some point. The basics of IA are generally agreed e.g., to add two intervals xx, yy, find an interval containing all x + y for x in xx and y in yy.

Many versions of IA theory exist, individually consistent but mutually incompatible. They differ especially in how to handle operations not everywhere defined on their inputs, such as division by an interval containing zero. In this situation a standard is called for, which not all will love but which is usable and practical in most IA applications.

The IEEE working group P1788, begun in 2008, has produced a draft standard for interval arithmetic, currently undergoing the IEEE approval process. The talk will concentrate on aspects of its architecture, especially:

- the levels structure, with a mathematical, a datum and an implementation level;

- the decoration system, which notes when a library operation is applied to input where it is discontinuous or undefined.

Time permitting, I may outline the P1788 flavor concept, by which implementations based on other versions of IA theory may be included into the standard in a consistent way.

##### 11 November 2014

**Speaker:** Natalia Kopteva (University of Limerick, Ireland).

**Title:** Maximum norm a posteriori error estimates for parabolic partial differential equations.

**Abstract:** Solutions of partial differential equations frequently exhibit corner singularities and/or sharp boundary and interior layers. To obtain reliable numerical approximations of such solutions in an efficient way, one may want to use meshes that are adapted to solution singularities. Such meshes can be constructed using a priori information on the solutions, however it is rarely available in real-life applications. Therefore the best hope for the future seems to be offered by the automated mesh construction by adaptive techniques. This approach requires no initial asymptotic understanding of the nature of the solutions and the solution singularity locations.

Reliable adaptive algorithms are based on a posteriori error estimates, i.e. estimates of the error in terms of values obtained in the computation process: computed solution and current mesh. Such a posteriori error estimates for parabolic partial differential equations will be the subject of this talk. For classical and singularly perturbed semilinear parabolic equations, we give computable a posteriori error estimates in the maximum norm, which, in the singularly perturbed regime, hold uniformly in the small perturbation parameter. The parabolic equations are discretized in time using the backward Euler, Crank-Nicolson and discontinuous Galerkin methods. Both semidiscrete (no spatial discretization) and fully discrete cases will be considered. The analysis invokes certain bounds for the Green's function of the parabolic operator. When dealing with the full discretizations, we also employ the elliptic reconstruction technique.

Although parts of our analysis are quite technical, it will be demonstrated (using a first-order ODE example as a trivial case of a parabolic PDE) that some main ideas are quite elementary.

##### 18 November 2014

**Speaker:** Robert Style (University of Oxford).

**Title:** The mechanics of soft solids - breaking classical laws.

**Abstract:** TBC.

Co-Host: Dr. Maurice Blount.

##### 25 November 2014

**Speaker:** Xuesong Wu (Imperial College London).

**Title:** TBC.

**Abstract:** TBC.

Co-Host: Dr. Chris Davies

##### 2 December 2014

**Speaker:** Matthias Heil (University of Manchester).

**Title:** Wrinkly fingers: fluid-structure interaction in elastic-walled Hele-Shaw cells.

**Abstract:** TBC.

Co-Host: Dr. Chris Davies

##### 3 February 2015

**Speaker:** Helen Wilson (University College London).

**Title:** Instabilities in viscoelastic fluids.

**Abstract:** TBC.

Co-Host: Professor Tim Phillips.

##### 10 February 2015

**Speaker:** Dmitri Tseluiko (University of Loughborough).

**Title:** TBC.

**Abstract:** TBC.

Co-Host: Dr Nikos Savva.

##### 17 February 2015

**Speaker:** David Needham (University of Birmingham).

**Title:** The evolution to localized and front solutions in a non-Lipschitz reaction-diffusion Cauchy problem with trivial initial data.

**Abstract:** TBC.

Co-Host: Professor Tim Phillips.

##### 17 March 2015

**Speaker:** Paul Milewski (University of Bath).

**Title:** Modelling and computation of pilot wave-bouncing droplet dynamics in a Faraday problem.

**Abstract:** TBC.

Co-Host: Dr Nikos Savva.

##### 24 March 2015

**Speaker:** Draga Pihler-Puzovi (University of Manchester).

**Title:** TBC.

**Abstract:** TBC.

Co-Host: Dr Maurice Blount.