Multilevel Preconditioner with Stable Coarse Grid Corrections for the Helmholtz Equation

In this paper we consider a class of robust multilevel preconditioners for the Helmholtz equation with high wave number. The key idea in this work is to use the continuous interior penalty finite element methods studied in [H. Wu, IMA J. Numer. Anal. , 34 (2014), pp. 1266--1288; L. Zhu and H. Wu, SIAM J. Numer. Anal. , 51 (2013), pp. 1828--1852] to construct the stable coarse grid correction problems. The multilevel methods, based on GMRES smoothing on coarse grids, are then served as a preconditioner in the outer GMRES iteration. In the one-dimensional case, the convergence property of the modified multilevel methods is analyzed by the local Fourier analysis. From our numerical results, we find that the proposed methods are efficient for a reasonable range of frequencies. The performance of the algorithms depends relatively mildly on wave number.