Recent research efforts aimed at iteratively solving time-harmonic waves have focused on a broad range of techniques to accelerate convergence. In particular, for the famous Helmholtz equation, deflation techniques have been studied to accelerate the convergence of Krylov subspace methods. In this work, we extend the two-level deflation method to a multilevel deflation method for (heterogeneous) Helmholtz and elastic wave problems. By using higher-order deflation vectors, we show that up to the level where the coarse-grid linear systems remain indefinite, the near-zero eigenvalues of the these coarse-grid operators remain aligned with the near-zero eigenvalues of the fine-grid operator, keeping the spectrum of the preconditioned system away from the origin. Combining this with the well-known CSLP-preconditioner, we obtain a scalable solver for the highly indefinite linear systems. This can be attributed to a close to wave number independent convergence and an optimal use of the CSLP-preconditioner on the indefinite levels. There, we approximate the CSLP-preconditioner, while allowing the complex shift to be small, by using inner Bi-CGSTAB iterations instead of a multigrid F-cycle. The proposed method shows very promising results for the more challenging two- and three-dimensional heterogeneous time-harmonic wave problems.

最近旨在迭代求解时间谐波波的研究工作集中在各种加速收敛的技术上。特别是对于著名的亥姆霍兹方程，已经研究了紧缩技术以加速 Krylov 子空间方法的收敛。在这项工作中，我们将两级通缩方法扩展到针对（异构）亥姆霍兹和弹性波问题的多级通缩方法。通过使用高阶紧缩向量，我们表明直到粗网格线性系统保持不确定的水平，这些粗网格算子的近零特征值与细网格的近零特征值保持一致算子，使预处理系统的频谱远离原点。将其与著名的 CSLP-preconditioner 相结合，我们为高度不确定的线性系统获得了可扩展的求解器。这可以归因于接近波数独立的收敛和在不确定水平上对 CSLP 预条件器的最佳使用。在那里，我们通过使用内部 Bi-CGSTAB 迭代而不是多重网格 F 循环来近似 CSLP 预处理器，同时允许复杂的偏移很小。对于更具挑战性的二维和三维异质时谐波问题，所提出的方法显示出非常有希望的结果。