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.
This approach relies on the fact that trigonometric polynomials are an orthonormal basis for
[2] The spectral element method chooses instead a high degree piecewise polynomial basis functions, also achieving a very high order of accuracy.
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 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.
[3] 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 Gauss-Lobatto quadrature using the same nodes.
The most popular applications of the method are in computational fluid dynamics[3] and modeling seismic wave propagation.
[4] The classic analysis of Galerkin methods and Céa's lemma holds here and it can be shown that, if
Similar results can be obtained to bound the error in stronger topologies.
[5] The LGL form of SEM is equivalent,[6] so it achieves the same superconvergence properties.
Development of the most popular LGL form of the method is normally attributed to Maday and Patera.
First, there is the Hybrid-Collocation-Galerkin method (HCGM),[8][5] which applies collocation at the interior Lobatto points and uses a Galerkin-like integral procedure at element interfaces.