We present an efficient matrix-free geometric multigrid method for the elastic Helmholtz equation, and a suitable discretization. Many discretization methods had been considered in the literature for the Helmholtz equations, as well as many solvers and preconditioners, some of which are adapted for the elastic version of the equation. However, there is very little work considering the reciprocity of discretization and a solver. In this work, we aim to bridge this gap. By choosing an appropriate stencil for re-discretization of the equation on the coarse grid, we develop a multigrid method that can be easily implemented as matrix-free, relying on stencils rather than sparse matrices. This is crucial for efficient implementation on modern hardware. Using two-grid local Fourier analysis, we validate the compatibility of our discretization with our solver, and tune a choice of weights for the stencil for which the convergence rate of the multigrid cycle is optimal. It results in a scalable multigrid preconditioner that can tackle large real-world 3D scenarios.

翻译：我们针对弹性亥姆霍兹方程提出了一种高效的无矩阵几何多重网格方法及合适的离散方案。文献中已广泛研究了亥姆霍兹方程的各种离散方法、求解器及预处理器，部分技术亦适用于该方程的弹性版本。然而，关于离散化与求解器之间互逆性的研究极为匮乏。本文旨在填补这一空白。通过选取合适的模板在粗网格上对方程进行重新离散，我们开发了一种易实现为无矩阵形式的多重网格方法——其依赖于模板而非稀疏矩阵。这对现代硬件上的高效实现至关重要。利用双网格局部傅里叶分析，我们验证了离散方案与求解器的兼容性，并调优得到一组最优收敛速率的模板权重系数。最终形成可扩展的多重网格预处理器，能够处理大规模真实三维场景。