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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 generally nonzero over the whole ___domain, while finite element methods use basis functions that are nonzero only on small subdomains ([[compact support]]). Consequently, spectral methods connect variables ''globally'' while finite elements do so ''locally''. 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'' increases is sometimes called a [[spectral-element method]].
Spectral methods can be used to solve [[differential equations]] (PDEs, ODEs, eigenvalue, etc){{Reflist}}<ref>{{cite journal |last1=Muradova |first1=Aliki D. |title=The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions |journal=Adv Comput Math |year=2008 |volume=29 |issue=2 |pages=179–206, 2008 |doi=10.1007/s10444-007-9050-7|hdl=1885/56758 |s2cid=46564029 |hdl-access=free }}</ref> and [[optimization problem]]s. When applying spectral methods to time-dependent PDEs, the solution is typically written as a sum of basis functions with time-dependent coefficients; substituting this in the PDE yields a system of ODEs in the coefficients which can be solved using any [[numerical methods for ordinary differential equations|numerical method for ODEs]]. Eigenvalue problems for ODEs are similarly converted to matrix eigenvalue problems {{Citation needed|date=August 2013}}.
Spectral methods were developed in a long series of papers by [[Steven Orszag]] starting in 1969 including, but not limited to, Fourier series methods for periodic geometry problems, polynomial spectral methods for finite and unbounded geometry problems, pseudospectral methods for highly nonlinear problems, and spectral iteration methods for fast solution of steady-state problems. The implementation of the spectral method is normally accomplished either with [[collocation method|collocation]] or a [[Galerkin method|Galerkin]] or a [[Tau method|Tau]] approach . For very small problems, the spectral method is unique in that solutions may be written out symbolically, yielding a practical alternative to series solutions for differential equations.
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