Vanka-smoothed shifted Laplacian multigrid preconditioners for the Helmholtz equations

TL;DR

The paper tackles the difficulty of solving high-frequency acoustic Helmholtz equations with scalable multigrid methods. It introduces a level-dependent intergrid scheme together with an additive Vanka smoother to keep the complex shift small while enabling deep V-cycles, and analyzes the approach with Local Fourier Analysis. A Galerkin coarse-grid correction, mixed-level intergrid operators, and carefully designed Vanka patches (Element, Plus, RB, Full) are combined with boundary treatment to achieve stable convergence. Numerical experiments in 2D and 3D, including heterogeneous geophysical media, show substantial runtime improvements over plain shifted Laplacian preconditioners and near-linear time scaling in 3D, validating both the method and its practical impact for large-scale Helmholtz problems.

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.

Figures (10)

• Figure 1: Patches for additive Vanka smoother for a 2D nodal discretization.
• Figure 1: LFA two-grid factor and convergence factor in practice for different damping parameters for each patch in 2D.
• Figure 2: Patches for additive Vanka smoother for a 3D nodal discretization.
• Figure 2: Numerically calculated relative residual and relative error, and asymptotic expected convergence rate predicted by LFA for each 2D patch.
• Figure 3: Magnitudes of the 2D Laplacian's stencil coefficients for different levels. The fine stencil is the compact 4-th order discretization described in \ref{['eq:disc4thCompact']}, and the other stencils are accepted by Galerkin coarse approximation with the level-dependent intergrid scheme described in \ref{['eq:levelDepIntergrid']}.

Theorems & Definitions (5)

• Definition 2.1: trottenberg2000multigrid, Chapter 8
• Definition 2.2: trottenberg2000multigrid, Chapter 8
• Definition 2.3: trottenberg2000multigrid, Chapter 8
• Remark
• Remark