A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number. (3rd May 2017)

• Record Type: Journal Article
• Authors: Du, Yu; Zhang, Zhimin
• Is Part Of: Communications in computational physics. Volume 22:Number 1(2017:Jul.)
• Page Start: 133
• Page End: 156
• Publication Date: 2017-05-03
• Subjects: 65N12, -- 65N15, -- 65N30, -- 78A40; Weak Galerkin finite element method, -- Helmholtz equation, -- large wave number, -- stability, -- error estimates; Mathematical physics -- Data processing -- Periodicals; Physics -- Data processing -- Periodicals; 530.150285
• DOI: 10.4208/cicp.OA-2016-0121
• Languages: English
• ISSNs: 1815-2406

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 2 ≤ C 0 or ( kh ) 2 + k ( kh ) p +1 ≤ C 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.