# Dispersive and dissipative behavior of the spectral element method

Mark Ainsworth, Hafiz Abdul Wajid

Research output: Contribution to journalArticle

### Abstract

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.
Language English 3910-3937 28 SIAM Journal on Numerical Analysis 47 5 16 Dec 2009 10.1137/080724976 Published - 2009

### Fingerprint

Spectral Element Method
Gauss Points
Finite Element
Wavelength
Mesh
Tend
Spectral Elements
Phase-lag
Legendre
Pi
Lagrange
Numerical integration
Gauss
Resolve
Vertex of a graph

### Keywords

• mass lumped scheme
• numerical dispersion
• spectral element method

### Cite this

Ainsworth, Mark ; Wajid, Hafiz Abdul. / Dispersive and dissipative behavior of the spectral element method. In: SIAM Journal on Numerical Analysis. 2009 ; Vol. 47, No. 5. pp. 3910-3937.
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Dispersive and dissipative behavior of the spectral element method. / Ainsworth, Mark; Wajid, Hafiz Abdul.

In: SIAM Journal on Numerical Analysis, Vol. 47, No. 5, 2009, p. 3910-3937.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Dispersive and dissipative behavior of the spectral element method

AU - Ainsworth, Mark

AU - Wajid, Hafiz Abdul

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

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KW - numerical dispersion

KW - spectral element method

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