On Edge Multiscale Space based Hybrid Schwarz Preconditioner for Helmholtz Problems with Large Wavenumbers

TL;DR

This work tackles the challenge of solving the Helmholtz equation at large wavenumbers $k$ by introducing a two-level, edge multiscale space based hybrid Schwarz preconditioner (EMs-HS). The method couples a one-level RAS-imp preconditioner with a coarse correction constructed from an edge multiscale space designed to approximate the global Helmholtz harmonic extension, enabling contraction of the preconditioned operator under mild mesh-resolutions and level-parameters. Theoretical results, proven via Schatz-type arguments, show uniform convergence for large $k$ and provide optimal coarse-space dimension scaling of $\mathcal{O}(k^d)$ in the ideal uniform-mesh case. Numerical experiments in 2D and 3D, including heterogeneous media, validate the sharpness of the theory and demonstrate improved robustness and efficiency over traditional one-level and polynomial coarse-space hybrids. The approach promises scalable solutions for high-frequency Helmholtz problems and offers directions for extension to time-harmonic Maxwell equations.

Abstract

In this work, we develop a novel hybrid Schwarz method, termed as edge multiscale space based hybrid Schwarz (EMs-HS), for solving the Helmholtz problem with large wavenumbers. The problem is discretized using $H^1$-conforming nodal finite element methods on meshes of size $h$ decreasing faster than $k^{-1}$ such that the discretization error remains bounded as the wavenumber $k$ increases. EMs-HS consists of a one-level Schwarz preconditioner (RAS-imp) and a coarse solver in a multiplicative way. The RAS-imp preconditioner solves local problems on overlapping subdomains with impedance boundary conditions in parallel, and combines the local solutions using partition of unity. The coarse space is an edge multiscale space proposed in [13]. The key idea is to first establish a local splitting of the solution over each subdomain by a local bubble part and local Helmholtz harmonic extension part, and then to derive a global splitting by means of the partition of unity. This facilitates representing the solution as the sum of a global bubble part and a global Helmholtz harmonic extension part. We prove that the EMs-HS preconditioner leads to a convergent fixed-point iteration uniformly for large wavenumbers, by rigorously analyzing the approximation properties of the coarse space to the global Helmholtz harmonic extension part and to the solution of the adjoint problem. Distinctly, the theoretical convergence analysis are valid in two extreme cases: using minimal overlapping size among subdomains (of order $h$), or using coarse spaces of optimal dimension (of magnitude $k^d$, where $d$ is the spatial dimension). We provide extensive numerical results on the sharpness of the theoretical findings and also demonstrate the method on challenging heterogeneous models.

Figures (5)

• Figure 1: An illustration of hierarchical bases for $\ell=0,1,2$.
• Figure 2: Grid points for $\ell=0,1,2$ over $\partial\Omega_i$.
• Figure 3: The solution of the homogeneous model with perforation, $k=209.4$.
• Figure 4: The velocity field and solution of the 2-d heterogeneous model.
• Figure 5: The velocity field and solution of the 3-d heterogeneous model.

Theorems & Definitions (17)

• Theorem 2.1: Properties of the sesquilinear form $a(\cdot,\cdot): V\times V\to \mathbb{C}$
• Theorem 2.2: Discrete inf-sup condition
• Remark 2.1
• Theorem 3.1: Local-global approximation
• proof
• Lemma 3.1
• proof
• Proposition 4.1: Approximation properties of the hierarchical space $V_{\ell}$
• proof
• Lemma 5.1: Approximation properties of the projection $\mathcal{P}_{i,\ell}$