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| | Hello, my name is Andrew and my wife doesn't like it at all. I've always cherished residing in Alaska. Invoicing is my profession. To climb is something she would by no means give up.<br><br>Feel free to surf to my blog post: accurate psychic readings ([http://c045.danah.co.kr/home/index.php?document_srl=1356970&mid=qna straight from the source]) |
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| In [[mathematics]], the '''spectral element method''' is a high order [[finite element method]].
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| Introduced in a 1984 paper<ref>A. T. Patera. A spectral element method for fluid dynamics - Laminar flow in a channel expansion. ''Journal of Computational Physics'', 54:468--488, 1984.</ref> 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..."
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| The spectral element method is an elegant formulation of the [[finite element method]] with a high degree piecewise polynomial basis.
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| == Discussion ==
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| The [[spectral method]] expands the solution in trigonometric series, a chief advantage being that the resulting method is of very high order. This approach relies on the fact that trigonometric polynomials are an [[orthonormal basis]] for <math>L^2(\Omega)</math>. The spectral element method chooses instead a high degree piecewise polynomial basis functions, also achieving a very high order of accuracy. The Spectral Element Method is a relatively new computational technique, where the element is defined using a higher order polynomial. Such polynomials are usually orthogonal Chebyshev polynomials or very high order Lobatto polynomials over non-uniformly spaced nodes. In SEM computational error decreases exponentially as the order of approximating polynomial, therefore a fast convergence of solution to the exact solution is realized with lesser degree of freedom of structure in comparison with FEM.
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| In structural health monitoring FEM can be used for detecting large flaws in the structure, but as the flaws' size is reduced there is a need to use a high frequency wave with very small wavelength. Therefore, the FEM mesh must be much finer, resulting in increased computational time and an inexact solution, hence SEM is used for detecting small flaws. By using SEM computational errors are avoided and computation time is decreased.
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| Non-uniformity of nodes helps to make the mass matrix diagonal, which saves time and memory and is also very useful for adopting Central Difference Method(CDM)
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| The disadvantages of SEM include modeling of complex geometry, compared to FEM.
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| == A-priori error estimate ==
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| 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, ''u<sub>N</sub>'' is the approximate solution and <math>u \in H^{s+1}(\Omega)</math>:
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| :<math>\|u-u_N\|_{H^1(\Omega)} \leqq C_s N^{-s} \| u \|_{H^{s+1}(\Omega)}</math> | |
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| 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]]:
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| :<math>\|u-u_N\|_{H^1(\Omega)} \leqq C \exp( - \gamma N )</math>
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| where <math>\gamma</math> depends only on <math>u</math>.
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| == Related methods ==
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| * G-NI or SEM-NI: these 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 [[Gaussian numerical integration]] is used instead of integrals in the definition of the bilinear form <math>a(\cdot,\cdot)</math> and in the functional <math>F</math>. These method are a family of [[Petrov–Galerkin method]]s their convergence is a consequence of [[Strang's lemma]].
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| * The spectral element method uses tensor product space spanned by nodal basis functions associated with [[Gaussian_quadrature#Gauss–Lobatto_rules|Gauss–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. 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 many fewer degrees of freedom.<ref>Barna Szabó and [[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''.
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| * The ''hp'' finite element method ([[hp-FEM]]) combines the advantages of the ''h'' and ''p'' refinements to obtain extremely fast, 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>
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| ==Notes==
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| <references/>
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| {{Numerical PDE}}
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| {{DEFAULTSORT:Spectral Element Method}}
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| [[Category:Numerical differential equations]]
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| [[Category:Partial differential equations]]
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| [[Category:Continuum mechanics]]
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| [[Category:Finite element method| ]]
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