In this work we extend the shifted Laplacian approach to the elastic Helmholtz equation. The shifted Laplacian multigrid method is a common preconditioning approach for the discretized acoustic Helmholtz equation. In some cases, like geophysical seismic imaging, one needs to consider the elastic Helmholtz equation, which is harder to solve: it is three times larger and contains a nullity-rich grad-div term. These properties make the solution of the equation more difficult for multigrid solvers. The key idea in this work is combining the shifted Laplacian with approaches for linear elasticity. We provide local Fourier analysis and numerical evidence that the convergence rate of our method is independent of the Poisson's ratio. Moreover, to better handle the problem size, we complement our multigrid method with the domain decomposition approach, which works in synergy with the local nature of the shifted Laplacian, so we enjoy the advantages of both methods without sacrificing performance. We demonstrate the efficiency of our solver on 2D and 3D problems in heterogeneous media.

翻译：本文将移位拉普拉斯方法拓展至弹性亥姆霍兹方程。移位拉普拉斯多重网格法是离散声学亥姆霍兹方程的一种常见预条件方法。在某些场景下（如地球物理地震成像），需考虑更难求解的弹性亥姆霍兹方程：其规模扩大三倍且包含富含零空间的梯度-散度项。这些特性使得多重网格求解器对该方程的求解更为困难。本文的核心思想是将移位拉普拉斯方法与线性弹性方法相结合。通过局部傅里叶分析与数值实验证明，本方法的收敛速率与泊松比无关。此外，为更好应对问题规模，我们将多重网格法与区域分解方法互补结合——后者与移位拉普拉斯法的局部特性协同运作，从而在不牺牲性能的前提下兼取两种方法的优势。最终在二维与三维非均匀介质问题中验证了求解器的有效性。