Large high-frequency free-space Helmholtz equations can be difficult to solve with the usual iterative solvers such as multigrid and domain decomposition A method to solve such problems is to solve them with a complex wave number as a preconditioner for a Helmholtz equation without damping, thereby modifying the equation for the preconditioner in such a way that the convergence rate is improved. This strategy is called the complex shifted Laplacian (CSL) (even though it is not the Laplacian that is shifted), and it uses the following method:

The preconditioner matrix for the CSL preconditioner Mβ is given by the discretization of the equation

where you choose a value of β such that you get evanescent wave solutions. Instead of solving the original system of equation, you now solve

You can activate complex shifted Laplacian for the following iterative solvers: Multigrid (GMG, and on multigrid levels for AMG), Domain Decomposition (Schur), and Domain Decomposition (Schwarz). For details, see the documentation for those solvers.