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In the numerical solution of partial differential equations, a topic in mathematics, the spectral element method (SEM) is a formulation of the finite element method (FEM) that uses high degree piecewise polynomials as basis functions. The spectral element method was introduced in a 1984 paper^{[1]} by A. T. Patera.
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 L^2(\Omega) . 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 Legendre 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 fewer degrees of freedom of the structure in comparison with FEM. In 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 with a small wavelength. Therefore, the FEM mesh must be much finer, resulting in increased computational time and an inexact solution. SEM, with fewer degrees of freedom per node, can be useful for detecting small flaws. Non-uniformity of nodes helps to make the mass matrix diagonal, which saves time and memory and is also useful for adopting central difference method (CDM). The disadvantages of SEM include modeling of complex geometry, compared to FEM.
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_{N} is the approximate solution and u \in H^{s+1}(\Omega):
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 \gamma depends only on u.
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