# Contents

## Idea

Spectral methods are methods for the numerical approximation of partial differential equations. They are important for the solution of the Navier-Stokes equations in meteorology and in climate models.

The basic idea of spectral methods is to choose a finite set of functions and calculate the optimal approximation of the exact solution by these functions. These basis functions are often part of an orthonormal basis of a Hilbert space, and more specifically trigonometric functions, hence the name “spectral” methods.

## Details

### Introduction for Pure Functional Analysts

The following paragraph is meant as an introduction to the method for pure mathematicians with a background in functional analysis.

For illustrative purposes we will make some simplifying assumptions. Let’s assume that we have an infinite topological vector space T, its topological dual $T^*$ and a (closable densely defined differential) operator

$A: D \subset T \to T$

with a unique solution of the equation

$A(f) = 0$

We omit initial and boundary conditions for the moment. In order to calculate an approximation to the exact solution $f$, we need to turn the infinite dimensional problem to a finite dimensional one.

The basic idea of spectral methods is to choose a finite dimensional subspace $T_g$ of T spanned by a given set of functions $\{g_1, ..., g_n \}$, which are called in this context trial, expansion or approximation functions. We are looking for the projection of the exact solution $f$ to the subspace $T_g$, but since we don’t know $f$, we cannot calculate the exact expansion

$f = \sum_{k = 1}^n \alpha_k g_k$

But we can test the goodness of a given approximation $f_{\alpha} := \sum_{k = 1}^n \alpha_k g_k$ by testing $M(f_{\alpha})$ for “smallness”.

The “smallness” test in spectral methods is done via a choice of a finite dimensional subspace $T^*_h$ of the dual space $T^*$ spanned by the elements $\{h_1, ..., h_n \}$, we then demand that

$\langle \; M(f_{\alpha}), \; h_i \; \rangle = 0$

should hold for all $h_i \in T^*_h$. The “functions” $h_i$ are called test or weight functions. Due to the choice of finite dimensional subspaces the problem is reduced to a finite set of (linear) algebraic equations.

### Approximation with Orthonormal Functions in Hilbert Space

Often the function space is a Hilbert space and the approximation functions elements of an orthonormal base. This is the case for example if one uses Fourier series. In some cases it is possible to show that the expansion coefficients of, for example, a smooth function in an orthonormal basis decay exponentially fast. This is sometimes called exponential or spectral accuracy in the literature.

## References

• Spectral methods, Wikipedia

• Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, Thomas A. Zang: Spectral methods. Fundamentals in single domains. (Springer 2006, ZMATH)

• Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, Thomas A. Zang: Spectral methods. Evolution to complex geometries and applications to fluid dynamics. (Springer 2007, ZMATH)

• David A. Kopriva: Implementing spectral methods for partial differential equations. Algorithms for scientists and engineers. (Springer 2009, ZMATH)