We present a matrix-free parallel scalable multilevel deflation preconditioned method for heterogeneous time-harmonic wave problems. Building on the higher-order deflation preconditioning proposed by Dwarka and Vuik (SIAM J. Sci. Comput. 42(2):A901-A928, 2020; J. Comput. Phys. 469:111327, 2022) for highly indefinite time-harmonic waves, we adapt these techniques for parallel implementation in the context of solving large-scale heterogeneous problems with minimal pollution error. Our proposed method integrates the Complex Shifted Laplacian preconditioner with deflation approaches. We employ higher-order deflation vectors and re-discretization schemes derived from the Galerkin coarsening approach for a matrix-free parallel implementation. We suggest a robust and efficient configuration of the matrix-free multilevel deflation method, which yields a close to wavenumber-independent convergence and good time efficiency. Numerical experiments demonstrate the effectiveness of our approach for increasingly complex model problems. The matrix-free implementation of the preconditioned Krylov subspace methods reduces memory consumption, and the parallel framework exhibits satisfactory parallel performance and weak parallel scalability. This work represents a significant step towards developing efficient, scalable, and parallel multilevel deflation preconditioning methods for large-scale real-world applications in wave propagation.

翻译：本文提出了一种面向异构时谐波问题的无矩阵并行可扩展多层消去预条件方法。该方法基于Dwarka和Vuik（SIAM J. Sci. Comput. 42(2):A901-A928, 2020; J. Comput. Phys. 469:111327, 2022）针对高度不定时谐波问题提出的高阶消去预条件技术，将其适配于并行实现，以求解具有最小污染误差的大规模异构问题。我们提出的方法将复平移拉普拉斯预条件子与消去方法相结合。我们采用源自Galerkin粗化方法的高阶消去向量和重新离散化方案，以实现无矩阵并行实现。我们提出了一种稳健高效的无矩阵多层消去方法配置，该方法能实现接近与波数无关的收敛性以及良好的时间效率。数值实验证明了该方法对于日益复杂的模型问题的有效性。预条件Krylov子空间方法的无矩阵实现降低了内存消耗，并行框架展现出令人满意的并行性能和弱并行可扩展性。这项工作为开发面向波传播大规模实际应用的高效、可扩展、并行的多层消去预条件方法迈出了重要一步。