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Spectral element method
In mathematics, the spectral element method is a high order finite element method.
Introduced in a 1984 paper by A. T. Patera, the abstract begins: "A spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques..."
The spectral element method is an elegant formulation of the finite element method with a high degree piecewise polynomial basis. The only relationship it has with the spectral method is its good convergence properties.
Additional recommended knowledge
The spectral method expands the solution in trigonometric series, a chief advantage is that the resulting method is of very high order. This approach relies on the fact that trigonometric polynomials are an orthonormal basis for L2(Ω). The spectral element method chooses instead high degree piecewise polynomial basis functions, also achieving a very high order of accuracy.
A-priori error estimate
The classic analysis of Galerkin methods and Céa's lemma holds here and it can be shown that, if u is the solution of the weak equation, uN is the approximate solution and :
where C is independent from N and s is no larger than the degree of the piecewise polynomial basis. As we increase N, we can also increase the degree of the basis functions. In this case, if u is an analytic function:
where γ depends only on u.
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Spectral_element_method". A list of authors is available in Wikipedia.|