Volume 22, Issue 1
A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number

Yu Du & Zhimin Zhang

Commun. Comput. Phys., 22 (2017), pp. 133-156.

Published online: 2019-10

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  • Abstract

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 k7/2h2 ≤C0 or (kh)2+k(kh)p+1 ≤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.

  • Keywords

Weak Galerkin finite element method, Helmholtz equation, large wave number, stability, error estimates.

  • AMS Subject Headings

65N12, 65N15, 65N30, 78A40

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

duyu87@csrc.ac.cn (Yu Du)

zmzhang@csrc.ac.cn (Zhimin Zhang)

  • BibTex
  • RIS
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@Article{CiCP-22-133, author = {Du , Yu and Zhang , Zhimin }, title = {A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number}, journal = {Communications in Computational Physics}, year = {2019}, volume = {22}, number = {1}, pages = {133--156}, abstract = {

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 k7/2h2 ≤C0 or (kh)2+k(kh)p+1 ≤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.

}, issn = {1991-7120}, doi = {https://doi.org/ 10.4208/cicp.OA-2016-0121}, url = {http://global-sci.org/intro/article_detail/cicp/13350.html} }
TY - JOUR T1 - A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number AU - Du , Yu AU - Zhang , Zhimin JO - Communications in Computational Physics VL - 1 SP - 133 EP - 156 PY - 2019 DA - 2019/10 SN - 22 DO - http://dor.org/ 10.4208/cicp.OA-2016-0121 UR - https://global-sci.org/intro/article_detail/cicp/13350.html KW - Weak Galerkin finite element method, Helmholtz equation, large wave number, stability, error estimates. AB -

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 k7/2h2 ≤C0 or (kh)2+k(kh)p+1 ≤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.

Yu Du & Zhimin Zhang. (2019). A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number. Communications in Computational Physics. 22 (1). 133-156. doi: 10.4208/cicp.OA-2016-0121
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