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Abstract
If the nodes for the spectral element method are chosen to be the GaussLegendreLobatto points and a Lagrange basis is used, then the resulting mass matrix is diagonal and the method is sometimes then described as the Gausspoint 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 meshsize and the order of the scheme. We prove that (a) the Gausspoint 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 meshsize $h$, 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.
Original language  English 

Pages (fromto)  39103937 
Number of pages  28 
Journal  SIAM Journal on Numerical Analysis 
Volume  47 
Issue number  5 
Early online date  16 Dec 2009 
DOIs  
Publication status  Published  2009 
Keywords
 mass lumped scheme
 numerical dispersion
 spectral element method
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Dive into the research topics of 'Dispersive and dissipative behavior of the spectral element method'. Together they form a unique fingerprint.Projects
 1 Finished

ADAPTIVE NUMERICAL METHODS FOR OPTOELECTRONIC DEVICES PFACT 69
Ainsworth, M., Mottram, N. & Ramage, A.
EPSRC (Engineering and Physical Sciences Research Council)
2/04/07 → 30/11/10
Project: Research