Numerical analysis of the Galerkin and weak Galerkin method for the Helmholtz equation with high wave number

Event date: Wednesday, November 2, 2016
Location: Clements Hall 126
Time: 3:30–4:30pm

Featured Speaker: Dr. Zhimin Zhang, Department of Mathematics, Wayne State University and the Beijing Center for Computational Science

Abstract: We study convergence property of the weak Galerkin method of fixed degree p and supercovergence property of the linear finite element method for the Helmholtz problem with large wave number.

1. Using a modified duality argument, we improve the existing error estimates of the WG method, in particular, the error estimates with explicit dependence on the wave number k are derived, it is shown that if k(kh)^p+1 is sufficiently small, then the pollution error in the energy norm is bounded by O(k(kh)^2p), which coincides with the phase error of the finite element method obtained by existent dispersion analyses.
2. For linear finite element method under certain mesh condition, we obtain the H¹-error estimate with explicit dependence on the wave number k and show that the error between the finite element solution and the linear interpolation of the exact solution is superconvergent in the H¹-seminorm, although the pollution error still exists. We proved a similar result for the recovered gradient by polynomial preserving recovery (PPR) and found that the PPR can only improve the interpolation error and has no effect on the pollution error. Furthermore, we estimated the error between the finite element gradient and recovered gradient and discovered that the pollution error is canceled between these two quantities. Finally, we apply the Richardson extrapolation to the recovered gradient and demonstrate numerically that PPR combined with the Richardson extrapolation can reduce the interpolation and pollution errors simultaneously, and therefore, leads to an asymptotically exact a posteriori error estimator.

All theoretical findings are verified by numerical tests. READ MORE