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

TL;DR

This paper introduces a two-level restricted additive Schwarz preconditioner for heterogeneous Helmholtz problems, leveraging multiscale spectral generalized FEM to achieve convergence rates independent of mesh size and subdomain count, and improving with higher wavenumber.

Contribution

The paper develops a novel two-level RAS preconditioner based on MS-GFEM, with proven convergence properties that are independent of mesh size and subdomain number, and decays with increasing wavenumber.

Findings

Convergence rate $ ext{Lambda}$ is independent of mesh size and subdomain count.

Preconditioner achieves exponential convergence with few iterations due to MS-GFEM oversampling.

Numerical experiments confirm effectiveness in 2D, 3D, and high-contrast scenarios.

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 $\Lambda$ under some reasonable conditions, where $\Lambda$ 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 $\Lambda$ 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 $h\sim k^{-1-\gamma}$ for some $\gamma\in (0,1]$, it holds that $\Lambda \sim k^{-1+\frac{\gamma}{2}}$. 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.