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：我们提出了一种适用于弹性亥姆霍兹方程的高效无矩阵几何多重网格方法及其相应的离散格式。文献中已有众多针对亥姆霍兹方程的离散方法、求解器及预条件子，其中部分方法也适用于弹性版本方程。然而，关于离散格式与求解器相互适配性的研究却极为匮乏。本研究旨在弥合这一空白。通过选取合适的粗网格重离散模板，我们构建了一种可便捷实现为无矩阵形式的多重网格方法——该方法依赖模板而非稀疏矩阵，这对于在当代硬件上的高效实现至关重要。借助两网格局部傅里叶分析，我们验证了离散格式与求解器的兼容性，并调优了模板的权重参数，使多重网格循环的收敛速度达到最优。最终形成的可扩展多重网格预条件子能够有效处理大规模真实三维场景。