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：本研究将移位拉普拉斯方法推广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常见的预条件处理手段。在地球物理地震成像等场景中，需考虑求解难度更高的弹性亥姆霍兹方程：其规模增大三倍，且包含具有丰富零空间的梯度-散度项。这些特性使得多重网格求解器难以有效处理该方程。本文的核心思想在于将移位拉普拉斯方法与线弹性问题的求解策略相结合。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，我们采用区域分解方法对多重网格法进行补充，该方法与移位拉普拉斯算子的局部特性协同作用，可在不牺牲性能的前提下兼得两种方法的优势。我们在二维及三维非均匀介质问题中验证了该求解器的有效性。