Boundary Element Methods

Unlike domain discretisation methods, such as the finite element method or the finite-difference method, the boundary element method is a powerful numerical method which is attractive mainly due to the possibility of reducing the dimensionality (by one) of a boundary value problem described by linear partial differential equations. This results in significant computational savings and computer storage requirements. However, to be successful in the reduction of dimensionality, the fundamental solution of the original partial differential equation needs to be available in an analytical and simple explicit form. If this is not the case, then of much interest is the development and computational implementation of recent variants of the boundary element method, such as the dual reciprocity, multiple reciprocity, analogous element, contour point, plane wave, fundamental solutions and meshless methods for solving both direct and inverse problems of mathematical physics. Interest centres on treating singularities and noisy information on the boundaries.

Research in the Polymers and Industrial Mathematics group focuses on the mechanics of polymers and other complex fluids, free-surface flows and inverse problems. We are also concerned with the development and implementation of novel numerical and computational solution methods for both ordinary and partial differential equations, from fundamental aspects (the theoretical analysis of numerical methods) to problem-specific aspects (the design, development and practical implementation of novel algorithms). Within the
polymer area, we conduct fundamental research into fluids that have a complex microstructure, such as polymer melts and solutions and colloidal dispersions. Our research combines methods from molecular physics and continuum mechanics to develop multiscale models that link together the microscale motion of individual molecules to the flow behaviour of the bulk material. An important class of industrial flow problems are those involving free surfaces, such as in inkjet printing, film coating and bubble growth in polymeric foams. We also work on a diverse range of inverse problems in heat transfer, porous media, fluid and solid mechanics, acoustics and medicine. This is a strongly interdisciplinary subject and much of our research involves collaborations with independent research groups in science and engineering departments both at Leeds and worldwide, as well as with industry.