2024/09/10 by Chupeng Ma, Ma, Chupeng, Christian Alber +4 · 3 citations
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Composite Material Mechanics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.2409.06533
openalex publication_date 2024/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present and analyze a two-level restricted additive Schwarz (RAS) preconditioner for heterogeneous Helmholtz problems, based on a multiscale spectral generalized finite element method (MS-GFEM) proposed in [C. Ma, C. Alber, and R. Scheichl, SIAM. J. Numer. Anal., 61 (2023), pp. 1546--1584]. The preconditioner uses local solves with impedance boundary conditions, and a global coarse solve based on the MS-GFEM approximation space constructed from local eigenproblems. It is derived by first formulating MS-GFEM as a Richardson iterative method, and without using an oversampling technique, reduces to the preconditioner recently proposed and analyzed in [Q. Hu and Z.Li, arXiv 2402.06905]. We prove that both the Richardson iterative method and the preconditioner used within GMRES converge at a rate of Λ under some reasonable conditions, where Λ denotes the error of the underlying MS-GFEM \rsapproximation. Notably, the convergence proof of GMRES does not rely on the `Elman theory'. An exponential convergence property of MS-GFEM, resulting from oversampling, ensures that only a few iterations are needed for convergence with a small coarse space. Moreover, the convergence rate Λ is not only independent of the fine-mesh size h and the number of subdomains, but decays with increasing wavenumber k. In particular, in the constant-coefficient case, with h∼ k-1-γ for some γ∈ (0,1], it holds that Λ∼ k^-1+\fracγ2. We present extensive numerical experiments to illustrate the performance of the preconditioner, including 2D and 3D benchmark geophysics tests, and a high-contrast coefficient example arising in applications.