Contact Details
| Room No. | L901 |
|---|---|
| Telephone | +44 (0)141 548 4535 |
| Fax | +44 (0)141 548 3345 |
| m.ainsworth@strath.ac.uk |
Research Interests
Numerical analysis of partial differential equations
My research is in the area of numerical analysis of partial differential equations arising in physical sciences. The name of the game is to take a cheap (and often nasty) initial approximation, then, by looking at where the accuracy is unacceptable, design a new approximation by adaptively feeding back information. By continuing in this way, it is possible to end up with a near optimal approximation. At the very least, this can save vast amounts of computer time, or may mean the difference between getting an answer or not in some cases.
Often, partial differential equations are an idealised mathematical model for a physical problem. It is even possible to adaptively control the error in the underlying mathematical model. This is the idea behind hierarchical modelling-another area of my research.
Research on adaptive numerical methods includes a wide range of possibilities from mathematical analysis of the methods, to the design of efficient data structures and algorithms for their implementation. For instance, research students supervised by me have completed (or will complete) PhD theses in the following areas:
- David Kay (PhD 1997) The p and hp version of the finite element method applied to a class of non-linear elliptic PDEs. David is now a lecturer at Oxford University.
- Mark Arnold (PhD 1998) Hierarchic Modelling of Separable Elliptic Boundary Value Problems on Thin Domains. Mark went on to work for the government's veterinary laboratory.
- Bill Senior (PhD 1999) Data-Structures and Implementation of an Adaptive hp-Finite Element Method. Bill now develops finite element codes for Shell in the Netherlands.
- Pat Coggins (PhD 2000) Mixed hp-Finite Element Method for Viscous Incompressible Fluid Flow. Pat went on to work for the Met Office.
- David Blacker (PhD 2005) Robust non-conforming finite element methods for nearly incompressible elasticity. David went on to work for Transco.
I currently supervise three research students.
Recent Publications
- An adaptive multi-scale computational modelling of Clare College Bridge
- Mihai LA and Ainsworth M
Computer Methods in Applied Mechanics and Engineering, 2009, 198, 1839-1847. - An adaptive multi-scale approach to the modelling of masonry structures
- Ainsworth M and Mihai LA
International Journal for Numerical Methods in Engineering, 2009, 78, 1135-1163. - Explicit polynomial preserving trace liftings on a triangle
- Ainsworth M and Demkowicz L
Mathematische Nachrichten, 2009, 282, 640-658. - Guaranteed computable error bounds for conforming and nonconforming finite element analyses in planar elasticity
- Mark Ainsworth and Richard Rankin
University of Strathclyde Mathematics Research Report, 2009, #22 - A finite element procedure for the determination of rigorous numerical enclosures on the limit load in the analysis of masonry and multi-body structures
- L. Angela Mihai and Mark Ainsworth
University of Strathclyde Mathematics Research Report, 2009, #13
View all of Publications.
Vacancies & Studentships
Adaptive High Order Finite Element Methods for Optoelectronic Devices
Reference No.: MA1
Supervisor: Professor Mark Ainsworth
Date Advertised: 6th February 2007
Please contact Professor Mark Ainsworth for further information.
The project will involve the mathematical analysis and implementation of a new class of powerful numerical methods known as discontinuous Galerkin finite element methods applied to the numerical solution of partial differential equations arising in the modelling of liquid crystal displays.
An interest in mathematical analysis and applications would be an ideal background for the project and, although useful, it would not be necessary to have studied courses in numerical analysis.
This project is unusual in that it forms part of a much larger research project in collaboration with Hewlett Packard Research Labs in Bristol, so you would benefit from joining a team based in the Mathematics Department and at Hewlett-Packard and would participate in research meetings.
One advantage is that funding is guaranteed and I would expect to make a firm offer at an early stage to a suitably qualified candidate.
Parallel numerical algorithms and software for partial differential equations
Reference No.: MA2
Supervisor: Professor Mark Ainsworth
Date Advertised: 11th March 2009
Please contact Professor Mark Ainsworth for further information.
This is a fully funded four year EPSRC studentship leading to the degree of PhD in the area of development of parallel high order finite element software for the numerical solution of partial differential equations.
The position would be suitable for students with an undergraduate or masters degree in computer science, mathematics, science or engineering who have some experience of programming in a high level language.
Experience of parallel programming will be an advantage, but is not required since the training will include one year to be spent at Edinburgh Parallel Computing Centre learning these skills.
During the main three year part of the studentship, you will be located in the Mathematics Department at Strathclyde University in Glasgow, working under the supervision of Professor Mark Ainsworth to whom specific enquiries should be addressed.
The studentship forms part of a recent Science & Innovation Award held jointly with Edinburgh and Herriott-Watt Universities.
EPSRC Doctoral Training Account Studentship: Optimal Design of Diffraction Gratings
Reference No.: DTG0901
Supervisor: Professor Mark Ainsworth
Date Advertised: 28th May 2009
Please contact Dr Oleg Davydov for further information.
The design of diffraction gratings with prescribed optical characteristics features in many industrial applications and in particular has applications to the liquid crystal displays currently being developed by Hewlett-Packard.
The main issues are to
- design a grating that achieves the desired optical performance (or comes within some tolerance), whilst
- minimising the cost of the materials or minimising the energy absorption of the grating.
The optical performance of the grating is governed by partial differential equations (Maxwell's equations or Helmholtz equations depending on the particular model being adopted) whose coefficients depend on the material.
These equations are generally not tractable via analytic approaches and have to be approximated numerically and the first part of the project will involve the studying and developing a computer code to approximate the equations.
The goal is then to optimise the optical performance. This part of the project will require the study of techniques from optimisation theory and the currently topical area of PDE constrained optimisation.
The final step consists of development and analysis of a computational algorithm whereby the computer is able to determine a particular design that minimises the cost whilst achieving the desired performance.
This project is part-funded by Hewlett-Packard and the student will have the opportunity to spend time visiting Hewlett-Packard Research Laboratories in Bristol each year where they will have the opportunity to experience research in an industrial environment and to see the practical applications of their research.