We present a novel block-preconditioner for the elastic Helmholtz equation, based on a reduction to acoustic Helmholtz equations. Both versions of the Helmholtz equations are challenging numerically. The elastic Helmholtz equation is larger, as a system of PDEs, and harder to solve due to its more complicated physics. It was recently suggested that the elastic Helmholtz equation can be reformulated as a generalized saddle-point system, opening the door to the current development. Utilizing the approximate commutativity of the underlying differential operators, we suggest a block-triangular preconditioner whose diagonal blocks are acoustic Helmholtz operators. Thus, we enable the solution of the elastic version using virtually any existing solver for the acoustic version as a black-box. We prove a sufficient condition for the convergence of our method, that sheds light on the long questioned role of the commutator in the convergence of approximate commutator preconditioners. We show scalability of our preconditioner with respect to the Poisson ratio and with respect to the grid size. We compare our approach, combined with multigrid solve of each block, to a recent monolithic multigrid method for the elastic Helmholtz equation. The block-acoustic multigrid achieves a lower computational cost for various heterogeneous media, and a significantly lower memory consumption, compared to the monolithic approach. It results in a fast solution method for wave propagation problems in challenging heterogeneous media in 2D and 3D.

翻译：本文提出了一种基于约化为声学Helmholtz方程的弹性Helmholtz方程新型块预条件子。两种Helmholtz方程在数值求解上均具有挑战性。弹性Helmholtz方程作为偏微分方程组规模更大，且因其更复杂的物理特性而更难求解。近期研究表明，弹性Helmholtz方程可重构为广义鞍点系统，这为当前研究提供了突破口。利用基础微分算子的近似可交换性，我们提出了一种对角块为声学Helmholtz算子的块三角预条件子。由此，我们能够将任意现有声学方程求解器作为黑盒，用于求解弹性方程版本。我们证明了该方法收敛的充分条件，这为长期存疑的交换子在近似交换子预条件子收敛中的作用提供了理论依据。实验表明，我们的预条件子关于泊松比和网格尺寸均具有良好的可扩展性。我们将结合多重网格求解各模块的方法，与近期提出的弹性Helmholtz方程整体多重网格法进行了对比。相较于整体式方法，块-声学多重网格法在各种非均匀介质中实现了更低的计算成本和显著降低的内存消耗。这为二维和三维复杂非均匀介质中的波传播问题提供了一种快速求解方法。