Two-level restricted additive Schwarz preconditioner based on multiscale spectral generalized FEM for heterogeneous Helmholtz problems - Ma, Chupeng; Alber, Christian; Scheichl, Robert; Zhang, Yongwei (Journal of scientific computing, 2025)
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 approximation. 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. In particular, in the constant-coefficient, non-trapping case, with $$h\sim k^{-1-γ }$$for some $$γ \in (0,1]$$, it holds that $$Λ \sim k^{-1+\frac{γ }{2}}$$, with the coarse-space dimension $$\sim k^{d}\log ^{d}(k)$$. 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.
Online veröffentlicht: 22. November 2025
Gesehen am 27.01.2026