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In this paper we are concerned with fast algorithms for the systems arisingfrom the plane wave discretizations for two-dimensional Helmholtz equationswith large wave numbers. We consider the plane wave weighted least squares(PWLS) method and the plane wave discontinuous Galerkin (PWDG) method. The maingoal of this paper is to construct multilevel parallel preconditioners forsolving the resulting Helmholtz systems. To this end, we first build amultilevel overlapping space decomposition for the plane wave discretizationspace based on a multilevel overlapping domain decomposition method. Then,corresponding to the space decomposition, we construct an additive multilevelpreconditioner for the underlying Helmholtz systems. Further, we design bothadditive and multiplicative multilevel preconditioners with smoothers, whichare different from the standard multigrid preconditioners. We apply theproposed multilevel preconditioners with a {\it constant} coarsest mesh size tosolve two dimensional Helmholtz systems generated by PWLS method or PWDGmethod, and we find that the new preconditioners possess nearly stableconvergence, i.e., the iteration counts of the preconditioned iterative methods(PCG or PGMRES) with the preconditioners increase very slowly when the wavenumber increases (and the fine mesh size decreases).

Novel multilevel preconditioners for the systems arising from plane wave discretization of Helmholtz equations with large wave numbers