Dispersive and Dissipative Behavior of the Spectral Element Method

If the nodes for the spectral element method are chosen to be the Gauss-Legendre-Lobatto points and a Lagrange basis is used, then the resulting mass matrix is diagonal and the method is sometimes then described as the Gauss-point mass lumped finite element scheme. We study the dispersive behavior of the scheme in detail and provide both a qualitative description of the nature of the dispersive and dissipative behavior of the scheme along with precise quantitative statements of the accuracy in terms of the mesh-size and the order of the scheme. We prove that (a) the Gauss-point mass lumped scheme (i.e., spectral element method) tends to exhibit phase lag whereas the (consistent) finite element scheme tends to exhibit phase lead; (b) the absolute accuracy of the spectral element scheme is $1/p$ times better than that of the finite element scheme despite the use of numerical integration; (c) when the order $p$, the mesh-size $h$, and the frequency of the wave $\omega$ satisfy $2p+1 \approx \omega h$ the true wave is essentially fully resolved. As a consequence, one obtains a proof of the general rule of thumb sometimes quoted in the context of spectral element methods: $\pi$ modes per wavelength are needed to resolve a wave.