Spectral element method

In mathematics, the spectral element method is a high order finite element method.

In modern applications spectral methods are a family of Galerkin methods in which $V_{h}$ is the finite subspace of $V$ made of the polynomials of degree $N$ over $\Omega$ if $\Omega$ is the domain of the equation.

The modern approach

The classical approach to spectral method is to expand the solution in trigonometric series and to find a relation for its Fourier coefficients. This approach relies on the fact that trigonometric polynomials are an orthonormal basis for $L^{2}(\Omega )$, so a natural generalization of this method is to chose another orthonormal basis. For example the basis obtained using Chebyshev polynomials or Legendre polynomials: these are orthogonal polynomials so if we introduce a weighted inner product on $L^{2}$ which makes it an Hilbert space they forms an orthonormal basis for this space. Chebishev polynomials where mostly used in the '80s due to the existence of an effective fast Fourier transform algorithm, while FFT for the Legendre case exists, but is quite unefficient. Nevertheless Legendre polynomials are now mostly used because they have constant weight function and, with the use of spectral element methods, there is no need to have very high polynomial degrees to obtain excellent approximations.

From an abstract point of view spectral methods are a family of Galerkin methods. For simplicity we will consider only the elliptic case. Let:

$\mathbb {Q} _{N}(\Omega )={\bigg \{}v(x)=\sum _{m_{1},m_{2},\ldots ,m_{d}=0}^{N}a_{m_{1},m_{2},\ldots ,m_{d}}x_{1}^{m_{1}}x_{2}^{m_{2}}\ldots x_{d}^{m_{d}}{\bigg \}}$

where $\Omega \subset \mathbb {R} ^{d}$. If the weak formulation of the problem is:

$?u\in V\ :\ a(u,v)=F(v)\qquad \forall v\in V$

a general spectral method is in the form:

$?u_{N}\in V_{N}\ :\ a(u_{N},v_{N})=F(v_{N})\qquad \forall v_{N}\in V_{N}$

where $V_{N}=\mathbb {Q} _{N}\cap V$ in the case of orthogonal polynomials or the linear span of the trigonometric functions if we use the Fourier basis.

Generalizations of the spectral methods

• 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 $a(\cdot ,\cdot )$ and in the functional $F$. These method are a family of Petrov-Galerkin methods their convergence is a consequence of the Strang's Lemma.