A hybrid shifted Laplacian multigrid and domain decomposition preconditioner for the elastic Helmholtz equations

TL;DR

This paper develops a new hybrid preconditioner combining shifted Laplacian multigrid and domain decomposition techniques to efficiently solve the elastic Helmholtz equations, which are more complex than acoustic cases.

Contribution

It extends the shifted Laplacian multigrid method to elastic Helmholtz equations and integrates domain decomposition for improved scalability and robustness.

Findings

Convergence rate is independent of Poisson's ratio.

Method is effective for 2D and 3D heterogeneous media.

Hybrid approach improves solver efficiency and robustness.

Abstract

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.