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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
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Year:    2017

Author:    Yu Du, Zhimin Zhang

Communications in Computational Physics, Vol. 22 (2017), Iss. 1 : pp. 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cicp.OA-2016-0121

Communications in Computational Physics, Vol. 22 (2017), Iss. 1 : pp. 133–156

Published online:    2017-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    24

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

Author Details

Yu Du

Zhimin Zhang

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