Scalable multi-level deflation preconditioning for highly indefinite time-harmonic waves

Recent research efforts aimed at iteratively solving time-harmonic waves have focused on a broad range of techniques to accelerate convergence. In particular, for the famous Helmholtz equation, deflation techniques have been studied to accelerate the convergence of Krylov subspace methods. In this work, we extend the two-level deflation method to a multilevel deflation method for (heterogeneous) Helmholtz and elastic wave problems. By using higher-order deflation vectors, we show that up to the level where the coarse-grid linear systems remain indefinite, the near-zero eigenvalues of the these coarse-grid operators remain aligned with the near-zero eigenvalues of the fine-grid operator, keeping the spectrum of the preconditioned system away from the origin. Combining this with the well-known CSLP-preconditioner, we obtain a scalable solver for the highly indefinite linear systems. This can be attributed to a close to wave number independent convergence and an optimal use of the CSLP-preconditioner on the indefinite levels. There, we approximate the CSLP-preconditioner, while allowing the complex shift to be small, by using inner Bi-CGSTAB iterations instead of a multigrid F-cycle. The proposed method shows very promising results for the more challenging two- and three-dimensional heterogeneous time-harmonic wave problems.