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：本文工作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化声学亥姆霍兹方程常用的预条件方法。在地球物理地震成像等某些应用场景中，需要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富含零空间的梯度-散度项。这些特性使得多重网格求解器求解该方程更为困难。本研究的核心思路是将移位拉普拉斯方法与线性弹性问题求解策略相结合。我们通过局部傅里叶分析与数值实验证明，该方法的收敛速率不随泊松比变化。此外，为更好应对问题规模，我们采用与移位拉普拉斯局部特性协同工作的域分解方法对多重网格进行补充，从而在保持计算性能的同时兼具两种方法的优势。最后，我们通过二维与三维非均匀介质算例验证了该求解器的有效性。