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Spectral method

Spectral methods are techniques used in applied mathematics and scientific computing to numerically solve certain differential equations, often involving the use of the fast Fourier transform. The idea is to write the solution of the differential equation as a sum of certain "basis functions" (for example, as a Fourier series, which is a sum of sinusoids) and then to choose the coefficients in the sum that best satisfy the differential equation.

Spectral methods and finite element methods 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. However, there are no known three-dimensional single domain spectral shock capturing results. 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.