SCAIM Seminar: Ben Adcock (SFU)

While spectral methods for the numerical solution of PDEs with smooth solutions offer the great advantage of high accuracy, they are typically poorly suited for solving problems with nonsmooth or sharply peaked solutions. This is in great part due to the appearance of Gibbs phenomenon, which both inhibits fast convergence and pollutes the approximation with spurious oscillations. Although numerous methods exist for overcoming this effect, most are either numerically ill-conditioned or do not deliver high global accuracy. The purpose of this talk is to describe a new, stable approach for this problem. As I will discuss, aside from being well-conditioned, the resulting method is also optimal in a certain sense and simple to implement.

The recovery of a function from its Fourier or orthogonal polynomial coefficients can be viewed as an example of an abstract reconstruction problem in a separable Hilbert space. Techniques from computational spectral theory can now be used to design a stable procedure. In the first part of this talk I will describe this general framework, before addressing the application to Fourier and orthogonal polynomial expansions in greater detail.