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)

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