Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation
Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation
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DOI：
10.1002/nme.1620382203
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发表时间：
1995-11
影响因子：
2.9
通讯作者：
F. Ihlenburg;I. Babuska
中科院分区：
文献类型：
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作者：
F. Ihlenburg;I. Babuska
当应用数值方法计算亥姆霍兹方程的驻波时，人们得到了在非色散介质中也是色散的“数值波”。数值波显示的相速度取决于亥姆霍兹方程的参数k。在色散分析中，研究了精确解和数值解之间的相位差。在本文中，作者最近的结果对一维问题的相位差进行了数值评估和讨论的其他工作针对这一主题的背景下。然后表明，以前的H1模误差估计具有非色散特征，但仅适用于极其精细网格上的中等或高波数。另一方面，最近证明的误差估计恒定分辨率包含污染项。在对精确解作某些假设的情况下，该项具有相位差的量级。因此，在色散分析的结果和数值分析的结果之间建立了联系。在整个文件中，介绍和讨论的理论结果是伴随着几个模型问题的数值评估。特别注意的性能的Galerkin方法与高阶多项式逼近p（H-P版本）。
When applying numerical methods for the computation of stationary waves from the Helmholtz equation, one obtains ‘numerical waves’ that are dispersive also in non-dispersive media. The numerical wave displays a phase velocity that depends on the parameter k of the Helmholtz equation. In dispersion analysis, the phase difference between the exact and the numerical solutions is investigated. In this paper, the authors' recent result on the phase difference for one-dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic. It is then shown that previous error estimates in H1-norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only. On the other hand, recently proven error estimates for constant resolution contain a pollution term. With certain assumptions on the exact solution, this term is of the order of the phase difference. Thus a link is established between the results of dispersion analysis and the results of numerical analysis. Throughout the paper, the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems. Special attention is given to the performance of the Galerkin method with a higher order of polynomial approximation p(h-p-version).