Two-level Restricted Additive Schwarz preconditioner based on Multiscale Spectral Generalized FEM for Heterogeneous Helmholtz Problems

TL;DR

This work develops a two-level Restricted Additive Schwarz preconditioner for heterogeneous Helmholtz problems by grounding it in Multiscale Spectral Generalized FEM (MS-GFEM). The approach yields a coarse space built from local GenEO-type eigenfunctions derived on oversampled subdomains, with convergence of both a Richardson iteration and GMRES governed by the MS-GFEM approximation error $\Lambda$, which decays with increasing wavenumber $k$ and benefits from oversampling through exponential eigenvalue decay. Under reasonable resolution and stability conditions, the method achieves $k$-robust performance and allows a small coarse space, and, in the constant-coefficient case with $h \sim k^{-1-\gamma}$, $\Lambda$ scales like $k^{-1+\gamma/2}$ (and $\Lambda \sim k^{-1}$ with a Dirichlet variant). Numerical experiments in 2D and 3D geophysical benchmarks (including Marmousi and Overthrust) demonstrate fast convergence, scalability with subdomain counts, and robustness to high-contrast coefficients, highlighting the method’s potential for seismic imaging with many sources. The paper also discusses practical aspects such as the cost of local eigenproblem solves and coarse-space factorization, and outlines future work toward multilevel extensions.

Abstract

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 \rs{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. 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 $h\sim k^{-1-γ}$ for some $γ\in (0,1]$, it holds that $Λ\sim k^{-1+\fracγ{2}}$. 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.

Figures (7)

• Figure 1: Illustration of a subdomain $\omega_i$ with the overlapping zone shown in color.
• Figure 2: Frequency Scaling test: the numbers of eigenfunctions $n_{min}$ needed to achieve the accuracy criterion \ref{['eq:accuracy_criterion']}for case (i) (left, with $H$ fixed) and case (ii) (right, with $kH$ fixed).
• Figure 3: The quantity $C_S$ for an interior and a boundary subdomains for linear (left) and quadratic (right) finite elements.
• Figure 4: Constant-coefficient example with $k=200$, $h=1/752$ and $m=16$: Residuals vs. iteration numbers for variable $n_{loc}$ (left, with $H/H^{\ast}=0.9$) and oversampling ratios (right, with $n_{loc}=25$).
• Figure 5: The velocity field of the Marmousi model.

Theorems & Definitions (26)

• Remark 2.3
• Remark 2.5
• Remark 2.6
• Remark 2.7
• Remark 2.8
• Remark 2.9
• Remark 2.11
• Lemma 2.12
• Lemma 2.13
• Remark 2.14