TY - JOUR T1 - A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number AU - Du , Yu AU - Zhang , Zhimin JO - Communications in Computational Physics VL - 1 SP - 133 EP - 156 PY - 2019 DA - 2019/10 SN - 22 DO - http://doi.org/ 10.4208/cicp.OA-2016-0121 UR - https://global-sci.org/intro/article_detail/cicp/13350.html KW - Weak Galerkin finite element method, Helmholtz equation, large wave number, stability, error estimates. AB -

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.