This paper presents a scalable multigrid preconditioner targeting large-scale systems arising from discontinuous Petrov-Galerkin (DPG) discretizations of high-frequency wave operators. This work is built on previously developed multigrid preconditioning techniques of Petrides and Demkowicz (Comput. Math. Appl. 87 (2021) pp. 12-26) and extends the convergence results form $\mathcal{O}(10^7)$ degrees of freedom (DOFs) to $\mathcal{O}(10^9)$ DOFs using a new scalable parallel MPI/OpenMP implementation. Novel contributions of this paper include an alternative definition of coarse-grid systems based on restriction of fine-grid operators, yielding superior convergence results. In the uniform refinement setting, a detailed convergence study is provided, demonstrating h and p robust convergence and linear dependence with respect to the wave frequency. The paper concludes with numerical results on hp-adaptive simulations including a large-scale seismic modeling benchmark problem with high material contrast.

翻译：本文针对高频波动算子的不连续Petrov-Galerkin（DPG）离散化所产生的大规模系统，提出了一种可扩展的多重网格预条件器。本工作基于Petrides与Demkowicz先前开发的多重网格预条件技术（Comput. Math. Appl. 87 (2021) pp. 12-26），并通过一种新的可扩展并行MPI/OpenMP实现，将收敛结果从$\mathcal{O}(10^7)$自由度（DOFs）推广至$\mathcal{O}(10^9)$自由度。本文的新贡献包括：基于细网格算子的限制操作定义了一种替代性的粗网格系统，从而获得了更优的收敛结果。在均匀加密设定下，我们提供了详细的收敛性研究，证明了h和p鲁棒收敛性以及与波动频率的线性相关性。本文最后给出了hp自适应模拟的数值结果，包括一个具有高材料对比度的大规模地震建模基准问题。