Spectral methods for PDEs

Abstract

[en] The main goal of is to indruce the basic techniques and results of numerical analysis for spectral methods. The required prerequisites to understand this text are basic knowledge of numerical methods, a introductory course on PDEs, on Real and Functional analysis, and on Sobolev spaces. For instance, chapters 1-2 from [Bre08] covers most of the necessary background on functional analysis and Sobolev spaces. A more thorough treatment of those topics can be found in [Eva22] or [BB11]. As a refresher, we will include the most important results in the Appendix. We will study the basics of spectral methods for elliptic boundary value problems and briefly discuss possible extension to more complicated geometries. Every method described will be implemented in MATLAB®. In finite element methods, the approximation spaces are usually piecewise polynomials with a fixed degree (normally linear or quadratic), and convergence is achieved by refining the mesh, in other words by $h \longrightarrow 0$ and convergence is usually $\mathcal{O}\left(h^2\right)$. Moreover the linear systems obtained by the discretizations are usually very large but sparse. The finite element method also has the advantage that it can be used for very complicated geometries.

In spectral methods, convergence is achieved by having one (or more) fixed domain and increasing the degree of polynomial approximation (by taking $N \longrightarrow \infty$ ). In this case, convergence depends on the regularity of the solution. A typical result is that convergence is $\mathcal{O}\left(N^{-k}\right)$ where the $k$ depends on the smoothness of the solution. When the solution is $C^{\infty}$, then convergence is spectrally fast : it is faster than $\mathcal{O}\left(N^{-k}\right)$ for any $k!$ Spectral methods are much faster when the solution is known to be very regular. In many applications where the solution is known to be very regular and high precision is needed, spectral methods are very useful. Spectral methods lead to dense matrices but since convergence is fast, good precision can be obtained with small matrices. However, spectral methods are not as efficient when the solutions are not that regular, and the extension to complicated geometries is more complicated to that of finite elements.

Other methods like the Spectral Element or Spectral - hp Finite element methods combine both approaches