A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number. (3rd May 2017) - Du, Yu; Zhang, Zhimin
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.