A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number
A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number
复制标题
高波数亥姆霍兹方程弱伽辽金法的数值分析
DOI：
10.4208/cicp.oa-2016-0121
复制
发表时间：
2017
影响因子：
3.7
通讯作者：
Zhimin Zhang
中科院分区：
文献类型：
--
作者：
Yu Du;Zhimin Zhang
本文研究了文献[24，38]中弱Galerkin有限元法（WG-FEM）对二维和三维大波数Helmholtz问题的误差分析。利用Zhu和Wu提出的一种修正的对偶性，我们得到了WG-FEM的预渐近误差估计。特别地，导出了与波数k有显式依赖关系的误差估计.这表明，在网格条件kh≤ C 0或（kh）+k（kh）≤ C 0下，破H1模的污染误差为O（k（kh）2 p），这与已有频散分析所得到的有限元法的相位误差一致.这里h是网格尺寸，p是近似空间的阶，C 0是与k和h无关的常数。此外，数值试验验证了理论结果，并说明了WG-FEM在减少污染影响方面的强大能力。AMS科目分类：65 N12、65 N15、65 N30、78 A40
We study the error analysis of the weak Galerkin finite element method in [24, 38] (WG-FEM) for the Helmholtz problem with large wave number in two and three dimensions. Using a modified duality argument proposed by Zhu and Wu, we obtain the pre-asymptotic error estimates of the WG-FEM. In particular, the error estimates with explicit dependence on the wave number k are derived. This shows that the pollution error in the broken H1-norm is bounded by O(k(kh)2p) under mesh condition kh≤C0 or (kh)+k(kh)≤C0, which coincides with the phase error of the finite element method obtained by existent dispersion analyses. Here h is the mesh size, p is the order of the approximation space and C0 is a constant independent of k and h. Furthermore, numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the WG-FEM in reducing the pollution effect. AMS subject classifications: 65N12, 65N15, 65N30, 78A40