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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.
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
with a unique solution of the equation
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
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
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.
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)