Liquid crystal displays have become an integral part of modern life. From the simple black and white display on a washing machine to enormous colour televisions, liquid crystal displays are used to provide a variety of information to the user. Whether that display tells you how long you have to wait until the washing is done or how the contestants in Big Brother are coping, there is an increasing demand for more and more sophisticated displays. Year after year the sales figures for liquid crystal displays increase and with many major electronics companies shutting down their CRT television factories to concentrate on liquid crystal display production, this proliferation of liquid crystal displays is not likely to slow in the near or mid-term future. While the performance of some displays is sufficient for a number of applications there are key market areas where advancements are needed. The ability to produce a high definition image is crucial for modern large screen televisions as well as in smaller portable displays where reducing the power used is also important. As televisions become larger (the largest liquid crystal television is now 70 ) and portable displays become more sophisticated, these issues are fuelling research into new liquid crystal display technologies. In particular, the ability to harness defects within the liquid crystal material has led to a number of novel displays which are multistable, giving them the ability to display an image with no power taken from the battery or power source. This will enable display manufacturers to produce more energy efficient, higher definition displays. There is however a problem. In order to fully develop such displays, to minimise the energy requirements and maximise the speed and resolution of the display, industrial companies often consider mathematical models of their displays so that many virtual experiments can be simulated numerically. The problem with these new types of displays is that, whilst the models have been formulated in terms of sets of differential equations, there are no accurate and robust numerical algorithms capable of approximating the equations. The differential equations, which model the fluid dynamics in complicated three dimensional regions as well as the motion of ions and the voltage through the display, are nonlinear and exhibit various localised phenomena including singularities that severely degrade the performance of standard numerical methods. In addition to these problems, to compare the model to experiments, the behaviour of light as it passes through the display must be considered and because of the complex structures found in these devices special numerical techniques must also be used. However, recent advances in the use of adaptive, high-order finite element methods have shown that such methods are accurate and robust in the sense that the performance does not degenerate when defects or abrupt changes in the structures exist. The main aim of this project is therefore to develop the theory necessary to realise robust and efficient high-order numerical methods for three-dimensional numerical simulation of liquid crystal display devices. The work will be undertaken in collaboration with Hewlett-Packard Laboratories in Bristol, who need these reliable numerical methods to take their new display technologies to the next level of development. It is particularly exciting that we will be able to immediately apply our new numerical approach to state-of-the-art industrial applications. These high-order finite element methods are a hot topic in current numerical analysis research and advances in this area would have important implications for solving partial differential equations in many other areas of science and engineering.