Numerical analysis and computation for partial differential equations on surfaces

This proposal is concerned with the numerical analysis and computation of partial differential equations on surfaces. There is a vast amount of research concerned with partial differential equations posed on domains in three space dimensions, say, but relatively very little on differential equations on hypersurfaces which may be the boundaries of bulk three dimensional domains. Such surface partial differential equations are of increasing importance in modelling complex surface processes in biology and materials. Traditionally equations on spheres arise in climate and weather modelling. Numerical methods in this setting can exploit the spherical geometry. However in the applications we have in mind, (eg cell biology, alloy surface dissolution, surfactants on fluid interfaces) the morpohology of the surface is arbitrary and may be complex. The challenges to be addressed in this project include:-mathematical analysis of degenerate equations related to discretization, hard numerical analysis, development of algorithms and application to complicated systems. The project is ambitious in scope because of the novelty and technicality of the mathematical problems and the scale of the applications. A PDRA will work closely with the PI on the adventurous complex problems and will receive advanced training in computational applied mathematics in an emerging and burgeoning area. Also involved will be two visiting researchers who are ongoing collaborators of the PI. The project will result in theorems concerning new methods, the development of algorithms and simulations of complex physical processes. Dissemination will be publication in high quality research articles and talks at conferences.