Professor Mark Ainsworth

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.