Spectral element method


In the numerical solution of partial differential equations, a topic in mathematics, the spectral element method is a formulation of the finite element method that uses high degree piecewise polynomials as basis functions. The spectral element method was introduced in a 1984 paper by A. T. Patera. Although Patera is credited with development of the method, his work was a rediscovery of an existing method

Discussion

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.
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 a central difference method.
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. The method gains its efficiency by placing the nodal points at the Legendre-Gauss-Lobatto points and performing the Galerkin method integrations with a reduced 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 and modeling seismic wave propagation.

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 depends only on.
The Hybrid-Collocation-Galerkin possesses some superconvergence properties. The LGL form of SEM is equivalent, 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. However, it was developed more than a decade earlier. First, there is the Hybrid-Collocation-Galerkin method, 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 is identical to SEM, while the HCGM is equivalent to these methods. This earlier work is ignored in the spectral literature.

Related methods