Solving differential equations with Fourier extension

Fourier series are a classical method for approximating functions which can be used in spectral methods to solve differential equations. However, Fourier series do not work well for non-periodic functions and they do not work at all on irregular domains in two or more dimensions.

Fourier extension provide a way around this problem. The idea is to extend the irregular domain to a rectangle and approximate the unknown function by a Fourier series defined on the rectangle. Earlier research shows that this method can in principle be used to solve differential equations but there are some practical issues. Firstly, Fourier series form an orthogonal basis leading to very nice properties, but Fourier extension does not form a basis which means that the approximation problem is ill conditioned. Secondly, the method needs to be sped up if we want to compete with existing methods; the basic idea here is to use the Fast Fourier Transform but the details are not clear.

The aim of this project is to design, analyze and implement a method based on Fourier extension for solving differential equations on irregular domains. We want a method with good performance for which we can prove spectral convergence. This project requires prior knowledge of numerical analysis (approximation theory, linear algebra, spectral methods) and the mathematical analysis that underpins this.