Scalable Multigrid Solver for the Helmholtz Equation: Real-Shifted Coarse Grid Correction
This work addresses the scalability bottleneck of multigrid solvers for high-frequency Helmholtz problems, which are critical in geophysical and acoustic simulations.
The authors present a scalable multigrid solver for high-frequency Helmholtz equations that achieves wavenumber-independent convergence without a complex shift, using a real-shifted coarse grid correction. For 12 grid points per wavelength, the method is scalable; for 10 grid points per wavelength, it outperforms the standard complex shifted Laplacian method by an order of magnitude.
We present a convergent and scalable multigrid solver for high-frequency Helmholtz equations. Standard multigrid methods do not converge for high-frequency Helmholtz problems, and a common cure is adding a complex shift and using the shifted operator as a preconditioner. Nevertheless, the complex shift prevents scalability. In this work we present a new method that achieves scalable convergence of a 3-level cycle without a complex shift. Our key idea is real-shifting the coarsest grid Galerkin operator, to correct the numerical dispersion between the grids. We show that this real-shifted coarse grid correction leads to a scalable 3-level method, for problems with 12 grid points per wavelength on the fine grid, and a convergent cycle with very few iterations for 11 grid points per wavelength, using standard point-smoothers. For problems with 10 grid points per wavelength, our method combined with a modest complex shift outperforms the standard complex shifted Laplacian method by an order of magnitude. We demonstrate wavenumber independent convergence for heterogeneous geophysical media in 2D and 3D.