A new level-dependent coarse grid correction scheme for indefinite Helmholtz problems

Pushed by the rising interest in high resolution requirements and high-dimensional applications, the diffusion term in the Laplacian equation drives the condition number of the associated discretized operator to undesirable sizes for standard iterative methods to converge rapidly. In addition, for realistic values of the wavenumber k(x) in (1), the Helmholtz operator H becomes indefinite, destroying the convergence behaviour of much preferred sparse linear system solvers such as e.g. Krylov subspace methods and classical geometric multigrid. When the negative shifting term −k(x) in the Helmholtz operator in (1) is replaced by a complex valued shift −(β1+ıβ2)k(x) the resulting operator is still closely related to the original, yet can efficiently be inverted with e.g. standard multigrid methods. This idea defined a well-known and successful Helmholtz preconditioning technique called complex shifted Laplacian [2]. The choice of the optimal value of the scaling parameter β1 + ıβ2 is a trade-off between a good preconditioner on the one hand and a computationally cheap inversion of that preconditioner on the other hand. Inspired by the complex shifted Laplacian, we present the construction and analysis of a modified multigrid method that is capable of solving the original indefinite Helmholtz equation (1) on the finest grid using a series of multigrid cycles with a level-dependent complex shift, i.e. gradually perturbing the original Helmholtz operator throughout the hierarchy, leading to a stable correction scheme on all levels.