A Numerical Analysis of the Weak Galerkin Method for the Helmholtz Equation with High Wave Number (Communications in Computational Physics, 2019)
We study the error analysis of the weak Galerkin finite element methodin [24, 38] (WG-FEM) for the Helmholtz problem with large wave number in two andthree dimensions. Using a modified duality argument proposed by Zhu and Wu, weobtain 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 thatthe 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 thefinite element method obtained by existent dispersion analyses. Here h is the meshsize, p is the order of the approximation space and C0 is a constant independent of kand h. Furthermore, numerical tests are provided to verify the theoretical findings andto illustrate the great capability of the WG-FEM in reducing the pollution effect.
Keywords: Weak Galerkin finite element method; Helmholtz equation; large wave number; stability; error estimates.