Turning redundancy into a computational advantage: frame-based discretisations of differential equations

Many processes in science and technology are described by mathematical models and equations. The equations capture the physical constraints of the process, and their solutions describe the actual outcomes. Sometimes these equations can be solved by hand, but in modern research more and more scientists rely on numerical computations with a fast computer. Most equations take the form of differential equations. These describe the way a system evolves or changes in response to inputs or environmental conditions. For the purpose of computer simulations, the solution is usually described by a sum of many basic functions. A common approach is to divide the domain into many small cells and use basic functions each concentrated locally around a single cell, called "finite elements". A superior approach, in terms of approximation power, is "spectral methods", which uses a globally supported set of basic functions with special properties. However, spectral methods so far have only been used for problems on simple geometries such as lines, rectangles, discs, spheres etc. This project proposes to use a set of basic functions which can be defined for complicated geometries, but are redundant (i.e. there are "too many" functions for a unique representation of a solution). It may appear at first that this redundancy will give an infeasible, "ill-conditioned" computational problem, but in this project we apply recent fundamental research which has been shown to circumvent the issue.