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In the numerical solution of partial differential equations, a topic in [[Mathematics|mathematics]], the '''spectral element method''' (SEM) is a formulation of the [[Finite element method|finite element method]] (FEM) that uses high degree [[Piecewise|piecewise]] [[Polynomial|polynomial]]s as basis functions. The spectral element method was introduced in a 1984 paper<ref>{{cite journal | first1=A. T. | last1=Patera | title=A spectral element method for fluid dynamics - Laminar flow in a channel expansion | journal=Journal of Computational Physics | volume=54 | number=3 | pages=468–488 | year=1984 | doi=10.1016/0021-9991(84)90128-1| bibcode=1984JCoPh..54..468P }}</ref> by A. T. Patera. Although Patera is credited with development of the method, his work was a rediscovery of an existing method (see Development History)
== Discussion ==

The [[Spectral method|spectral method]] expands the solution in [[Trigonometric polynomial|trigonometric]] series, a chief advantage being that the resulting method is of a very high order. 
This approach relies on the fact that [[Trigonometric polynomial|trigonometric polynomial]]s are an [[Orthonormal basis|orthonormal basis]] for <math>L^2(\Omega)</math>.<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> 
The spectral element method chooses instead a high degree piecewise polynomial basis functions, also achieving a very high order of accuracy. 
Such polynomials are usually orthogonal Chebyshev polynomials or very high order [[Lagrange polynomial|Lagrange polynomials]] over non-uniformly spaced nodes. 
In SEM computational error decreases exponentially as the order of approximating polynomial increases, therefore a fast convergence of solution to the exact solution is realized with fewer degrees of freedom of the structure in comparison with FEM.
In [[Engineering:Structural health monitoring|structural health monitoring]], FEM can be used for detecting large flaws in a structure, but as the size of the flaw is reduced there is a need to use a high-frequency wave. In order to simulate the propagation of a high-frequency wave, the FEM mesh required is very fine resulting in increased computational time. On the other hand, SEM provides good accuracy with fewer degrees of freedom. 
Non-uniformity of nodes helps to make the mass matrix diagonal, which saves time and memory and is also useful for adopting a central difference method (CDM). 
The disadvantages of SEM include difficulty in modeling complex geometry, compared to the flexibility of FEM.

Although the method can be applied with a modal piecewise orthogonal polynomial basis, it is most often implemented with a nodal tensor product Lagrange basis.<ref name=":0">Karniadakis, G. and Sherwin, S.: Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford Univ. Press, (2013), {{ISBN|9780199671366}}</ref> The method gains its efficiency by placing the nodal points at the Legendre-Gauss-Lobatto (LGL) points and performing the Galerkin method integrations with a reduced [[Gaussian quadrature|Gauss-Lobatto quadrature]] using the same nodes. With this combination, simplifications result such that mass lumping occurs at all nodes and a collocation procedure results at interior points.

The most popular applications of the method are in computational fluid dynamics<ref name=":0" /> and modeling seismic wave propagation.<ref>Komatitsch, D. and Villote, J.-P.: “The Spectral Element Method: An Efficient Tool to Simulate the Seismic Response of 2D and 3D Geologic Structures,” Bull. Seismological Soc. America, 88, 2, 368-392 (1998)</ref>

== A-priori error estimate ==
The classic analysis of [[Galerkin method]]s and [[Céa's lemma]] holds here and it can be shown that, if <math>u</math> is the solution of the weak equation, <math>u_N</math> is the approximate solution and <math>u \in H^{s+1}(\Omega)</math>:

:<math>\|u-u_N\|_{H^1(\Omega)} \leqq C_s N^{-s} \| u \|_{H^{s+1}(\Omega)}</math>

where <math>N</math> is related to the discretization of the domain (ie. element length), <math>C_s</math> is independent from <math>N</math>, and <math>s</math> is no larger than the degree of the piecewise polynomial basis. Similar results can be obtained to bound the error in stronger topologies. If <math>k \leq s+1</math>

<math>\|u-u_N \|_{H^k(\Omega)} \leq C_{s,k} N^{k-1-s} \|u\|_{H^{s+1}(\Omega)}</math>

As we increase <math>N</math>, we can also increase the degree of the basis functions. In this case, if <math>u</math> is an [[Analytic function|analytic function]]:

:<math>\|u-u_N\|_{H^1(\Omega)} \leqq C \exp( - \gamma N )</math>

where <math>\gamma</math> depends only on <math>u</math>.

The Hybrid-Collocation-Galerkin  possesses some superconvergence properties.<ref name=":2" /> The LGL form of SEM is equivalent,<ref name=":1">Young, L.C., “Orthogonal Collocation Revisited,” Comp. Methods in Appl. Mech. and Engr. 345 (1) 1033-1076 (Mar. 2019), [https://doi.org/10.1016/j.cma.2018.10.019 doi.org/10.1016/j.cma.2018.10.019]</ref> so it achieves the same superconvergence properties.

== Development History ==
Development of the most popular LGL form of the method is normally attributed to Maday and Patera.<ref>Maday, Y. and Patera, A. T., “Spectral Element Methods for the Incompressible Navier-Stokes Equations” In State-of-the-Art Surveys on Computational Mechanics, A.K. Noor, editor, ASME, New York (1989).</ref> However, it was developed more than a decade earlier. First, there is the Hybrid-Collocation-Galerkin method (HCGM),<ref>Diaz, J., “A Collocation-Galerkin Method for the Two-point Boundary Value Problem Using Continuous Piecewise Polynomial Spaces,” SIAM J. Num. Anal., 14 (5) 844-858 (1977) ISSN 0036-1429</ref><ref name=":2">Wheeler, M.F.: “A C0-Collocation-Finite Element Method for Two-Point Boundary Value and One Space Dimension Parabolic Problems,” SIAM J. Numer. Anal., 14, 1, 71-90 (1977)</ref> which applies collocation at the interior Lobatto points and uses a Galerkin-like integral procedure at element interfaces. The Lobatto-Galerkin method described by Young<ref>Young, L.C., “A Finite-Element Method for Reservoir Simulation,” Soc. Petr. Engrs. J. 21(1) 115-128, (Feb. 1981), paper SPE 7413 presented Oct. 1978, [https://doi.org/10.2118/7413-PA doi.org/10.2118/7413-PA]</ref> is identical to SEM, while the HCGM is equivalent to these methods.<ref name=":1" /> This earlier work is ignored in the spectral literature.

== Related methods ==
* G-NI or SEM-NI are the most used spectral methods. The Galerkin formulation of spectral methods or spectral element methods, for G-NI or SEM-NI respectively, is modified and Gauss-Lobatto integration is used instead of integrals in the definition of the [[Bilinear form|bilinear form]] <math>a(\cdot,\cdot)</math> and in the functional <math>F</math>.  Their convergence is a consequence of Strang's lemma.
*SEM is a Galerkin based FEM (finite element method) with Lagrange basis (shape) functions and reduced numerical integration by [[Gaussian quadrature|Lobatto quadrature]] using the same nodes.
*The [[Pseudo-spectral method|pseudospectral method]], [[Orthogonal collocation|orthogonal collocation]], differential quadrature method, and G-NI are different names for the same method. These methods employ global rather than piecewise polynomial basis functions. The extension to a piecewise FEM or SEM basis is almost trivial.<ref name=":1" />
* The spectral element method uses a [[Tensor product|tensor product]] space spanned by nodal basis functions associated with [[Gaussian quadrature#Gauss–Lobatto rules|Gauss&ndash;Lobatto point]]s. In contrast, the [[Hp-FEM|p-version finite element method]] spans a space of high order polynomials by nodeless basis functions, chosen approximately orthogonal for [[Numerical stability|numerical stability]]. Since not all interior basis functions need to be present, the p-version finite element method can create a space that contains all polynomials up to a given degree with fewer degrees of freedom.<ref>Barna Szabó and [[Biography:Ivo Babuška|Ivo Babuška]], Finite element analysis, John Wiley & Sons, Inc., New York, 1991. {{isbn|0-471-50273-1}}</ref>  However, some speedup techniques possible in spectral methods due to their tensor-product character are no longer available. The name ''p-version'' means that accuracy is increased by increasing the order of the approximating polynomials (thus, ''p'') rather than decreasing the mesh size, ''h''.
* The ''hp'' finite element method ([[Hp-FEM|hp-FEM]]) combines the advantages of the ''h'' and ''p'' refinements to obtain exponential convergence rates.<ref>P. Šolín, K. Segeth, I. Doležel: Higher-order finite element methods, Chapman & Hall/CRC Press, 2003. {{isbn|1-58488-438-X}}</ref>

==Notes==
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{{Numerical PDE}}

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[[Category:Numerical differential equations]]
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