A multigrid-based shifted Laplacian preconditioner for a fourth-order Helmholtz discretization

In this paper, an iterative solution method for a fourth-order accurate discretization of the Helmholtz equation is presented. The method is a generalization of that presented in (SIAM J. Sci. Comput. 2006; 27:1471-1492), where multigrid was employed as a preconditioner for a Krylov subspace iterative method. The multigrid preconditioner is based on the solution of a second Helmholtz operator with a complex-valued shift. In particular, we compare preconditioners based on a point-wise Jacobi smoother with those using an ILU(0) smoother, we compare using the prolongation operator developed by de Zeeuw in (J. Comput. Appl. Math. 1990; 33:1-27) with interpolation operators based on algebraic multigrid principles, and we compare the performance of the Krylov subspace method Bi-conjugate gradient stabilized with the recently introduced induced dimension reduction method, IDR(s). These three improvements are combined to yield an efficient solver for heterogeneous problems.