Dispersive and Dissipative Behavior of the Spectral Element Method
Dispersive and Dissipative Behavior of the Spectral Element Method
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DOI：
10.1137/080724976
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发表时间：
2009-10
期刊：
影响因子：
--
通讯作者：
M. Ainsworth;H. Wajid
中科院分区：
文献类型：
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作者：
M. Ainsworth;H. Wajid
如果谱元法的节点被选择为高斯-勒让德-洛巴托点，并且使用拉格朗日基，则所得的质量矩阵是对角的，并且该方法有时被描述为高斯点质量集中有限元格式。我们详细研究了该方案的色散行为，并提供了一个定性的描述的性质的色散和耗散行为的计划沿着精确的定量报表的准确性方面的网格大小和顺序的计划。我们证明了（a）高斯点质量集总方案（即，谱元素方法）倾向于表现出相位滞后，（B）尽管采用了数值积分，谱元格式的绝对精度仍比有限元格式高1/p 5倍;（c）当阶数p、网格尺寸h和波的频率满足2 p +1 \approx \omega h时，真正的波基本上是完全解析的。因此，人们得到了一个一般的经验法则的证明，有时引用的背景下，光谱元素的方法：$\pi$模式每波长需要解决一个波。
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.