Spectral method: Difference between revisions

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{{More footnotes|date=August 2013}}

'''Spectral methods''' are a class of techniques used in [[applied mathematics]] and [[scientific computing]] to numerically solve certain [[differential equation]]s, potentially involving the use of the [[Fastfast Fourier Transformtransform]]. The idea is to write the solution of the differential equation as a sum of certain "[[basis function|basis functions]]" (for example, as a [[Fourier series]] which is a sum of [[Sine wave|sinusoid]]s) and then to choose the coefficients in the sum in order to satisfy the differential equation as well as possible.

Spectral methods and [[finite element method]]s are closely related and built on the same ideas; the main difference between them is that spectral methods use basis functions that are nonzero over the whole ___domain, while finite element methods use basis functions that are nonzero only on small subdomains. In other words, spectral methods take on a ''global approach'' while finite element methods use a ''local approach''. Partially for this reason, spectral methods have excellent error properties, with the so-called "exponential convergence" being the fastest possible, when the solution is [[Smooth function|smooth]]. However, there are no known three-dimensional single ___domain spectral [[shock capturing]] results (shock waves are not smooth).<ref name="CHQZ">[https://books.google.com/books?id=7COgEw5_EBQC pp 235, Spectral Methods]: evolution to complex geometries and applications to fluid dynamics, By Canuto, Hussaini, Quarteroni and Zang, Springer, 2007.</ref> In the finite element community, a method where the degree of the elements is very high or increases as the grid parameter ''h'' decreases to zero is sometimes called a [[spectral element method]].