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

Part of: Partial differential equations, boundary value problems Optics, electromagnetic theory - General

Published online by Cambridge University Press:  03 May 2017

Yu Du  and
Zhimin Zhang
Yu Du*
Affiliation:
Beijing computational science research center, Beijing 100193, P.R. China
Zhimin Zhang*
Affiliation:
Beijing computational science research center, Beijing 100193, P.R. China Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
*
*Corresponding author. Email addresses:duyu87@csrc.ac.cn, dynju@qq.com (Y. Du), zmzhang@csrc.ac.cn, zzhang@math.wayne.edu (Z. Zhang)
*Corresponding author. Email addresses:duyu87@csrc.ac.cn, dynju@qq.com (Y. Du), zmzhang@csrc.ac.cn, zzhang@math.wayne.edu (Z. Zhang)
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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 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 methodHelmholtz equationlarge wave numberstabilityerror estimates

MSC classification

Secondary: 65N12: Stability and convergence of numerical methods 65N15: Error bounds 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 78A40: Waves and radiation
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 1 , July 2017 , pp. 133 - 156
Copyright
Copyright © Global-Science Press 2017 

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