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A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number

  • Yu Du (a1) and Zhimin Zhang (a1) (a2)
Abstract
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 H 1-norm is bounded by under mesh condition k 7/2 h 2C 0 or (kh)2+k(kh) p+1C 0, 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 C 0 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.

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Corresponding author
*Corresponding author. Email addresses: duyu87@csrc.ac.cn, dynju@qq.com (Y. Du), zmzhang@csrc.ac.cn, zzhang@math.wayne.edu (Z. Zhang)
References
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
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