Spectral Element vs Finite Element

Question

I am trying to understand the difference between SEM and FEM. If I go by this paper, spectral element methods are a subset of FEM methods and the only difference lies in the choice of basis functions. If this is the case, are there any advantages in using traditional FEM based Lagrange basis functions or SEM based on GLL Lagrange basis functions as this leads to dense matrices and bad condition numbers. In general, when would one prefer FEM over SEM if looking for high order methods?

Are there any open source libraries (ex. dealii, firedrake, fenics) that has SEM as a feature with all the common basis functions used in SEM (chebyshev, Legendre Galerkin or Lagrange basis using Gauss-Legendre-Lobatto or Gauss-Chebyshev Lobatto points).

Answer 1

The main advantage is that it reduces the Runge phenomenon and leads to faster convergence rates.

It also presents less numerical dispersion and need less nodes per wavelength (see 1 and 2). So, I would say that you would prefer the method for wave propagation scenarios.

Regarding software that includes SEM, I am aware of the following:

• FSELib: Matlab software accompanying the book Introduction to Finite and Spectral Element Methods using Matlab.
• Nektar: Spectral/HP FEM open code.
• RegSEM: used for seismic waves mainly.
• SPECFEM3D: used for seismic waves mainly.

References

1. Ainsworth, Mark, and Hafiz Abdul Wajid. "Dispersive and dissipative behavior of the spectral element method." SIAM Journal on Numerical Analysis 47.5 (2009): 3910-3937.

2. Guarín-Zapata, Nicolás, and Juan Gomez. "Evaluation of the spectral finite element method with the theory of phononic crystals." Journal of Computational Acoustics 23.02 (2015): 1550004.

Answer 2

The SEM is a FEM! It's almost like all these different names are designed to confuse the newcomer. I will speak primarily about the most popular form which uses a tensor product Lagrange basis with nodes at the Legendre-Gauss-Lobatto (LGL) points. Once you have chosen that basis and integrate with the LGL quadrature using the same nodes. You have a method with some nice computational features - lumped mass matrix, no integrals to evaluate, no interpolations needed to evaluate coefficients, can use conventional isoparametric elements, computational molecules with star structure for rectangular grids. In fact, it is a collocation method. You will find additional information in a recent article I wrote at https://doi.org/10.1016/j.cma.2018.10.019. I found the Wikipedia page (https://en.wikipedia.org/wiki/Spectral_element_method) so misleading I edited it recently, contradicting much of what was there.

Answer 3

As the name suggests, spectral element method is a combination of two methods: spectral collocation methods and finite element methods. The former has the highest convergence rate among all numerical methods, but normally can only be used for simple domains (a line, plane layers etc), so combining this with finite element methods makes it possible for spectral collocation methods to deal with complex geometries (since spectral collocation is still done on a simple parent element).

Spectral element method is particularly useful for wave propagation (electromagnetic, seismic etc.) simulations. For this group of problems, when appropriate techniques are used (i.e. Lagrange polynomials as interpolating function on GLL points), the most expensive step in the computation (i.e. the inversion of the mass matrix) becomes trivial since the mass matrix is diagonal. This I believe is the main advantage for using SEM for wave propagations. See e.g. Komatitsch and Tromp, 1999 for details.