Vanka-smoothed shifted Laplacian multigrid preconditioners for the Helmholtz equations

Abstract: We present an improved multigrid preconditioner for the acoustic Helmholtz equation with enhanced scalability. Standard multigrid fails to converge for the Helmholtz equation, and the well-known complex shifted Laplacian method overcomes it by adding a complex shift and using the shifted system as a preconditioner. However, the added complex shift grows with the frequency and interferes with the preconditioner's scalability. In this work, we present an additive Vanka smoother that requires a much lower shift than point-wise smoothers, and thereby enhances the scalability. By carefully designing different ingredients of the multigrid cycle, the presented method enables deep V-cycles with a small and bounded shift, even when many levels are used. We validate our method theoretically by local Fourier analysis, and hold numerical experiments for homogeneous and heterogeneous media. We show that our method outperforms plain shifted Laplacian in terms of runtimes and performs well on challenging geophysical media in 2D and 3D.