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Two-level hybrid Schwarz Preconditioners for The Helmholtz Equation with high wave number

Peipei Lu, Xuejun Xu, Bowen Zheng, Jun Zou

TL;DR

This work develops two-level hybrid Schwarz preconditioners for the Helmholtz equation at high wave number, where the coarse space is constructed via Localized Orthogonal Decomposition (LOD) and the subdomain solves use Dirichlet or impedance conditions. The authors prove uniform bounds on the preconditioned operator norm and the field of values, showing optimality that is independent of the fine and coarse mesh sizes, subdomain size, and wave number under reasonable assumptions. They also demonstrate exponential decay of the localized corrections and provide extensive numerical experiments validating robustness across a range of $h$, $H$, $H_{sub}$, $\\delta$, and $\\kappa$, with small oversampling ($m$) often sufficing. The practical impact is a scalable, robust preconditioning framework for GMRES on large, high-frequency Helmholtz problems, with coarse solvers that are simple, local, and decoupled from the subdomain details.

Abstract

In this work, we propose and analyze two two-level hybrid Schwarz preconditioners for solving the Helmholtz equation with high wave number in two and three dimensions. Both preconditioners are defined over a set of overlapping subdomains, with each preconditioner formed by a global coarse solver and one local solver on each subdomain. The global coarse solver is based on the localized orthogonal decomposition (LOD) technique, which was proposed in [30,31] originally for the discretization schemes for elliptic multiscale problems with heterogeneous and highly oscillating coefficients and Helmholtz problems with high wave number to eliminate the pollution effect. The local subproblems are Helmholtz problems in subdomains with homogeneous boundary conditions (the first preconditioner) or impedance boundary conditions (the second preconditioner). Both preconditioners are shown to be optimal under some reasonable conditions, that is, a uniform upper bound of the preconditioned operator norm and a uniform lower bound of the field of values are established in terms of all the key parameters, such as the fine mesh size, the coarse mesh size, the subdomain size and the wave numbers. It is the first time to show that the LOD solver can be a very effective coarse solver when it is used appropriately in the Schwarz method with multiple overlapping subdomains. Numerical experiments are presented to confirm the optimality and efficiency of the two proposed preconditioners.

Two-level hybrid Schwarz Preconditioners for The Helmholtz Equation with high wave number

TL;DR

This work develops two-level hybrid Schwarz preconditioners for the Helmholtz equation at high wave number, where the coarse space is constructed via Localized Orthogonal Decomposition (LOD) and the subdomain solves use Dirichlet or impedance conditions. The authors prove uniform bounds on the preconditioned operator norm and the field of values, showing optimality that is independent of the fine and coarse mesh sizes, subdomain size, and wave number under reasonable assumptions. They also demonstrate exponential decay of the localized corrections and provide extensive numerical experiments validating robustness across a range of , , , , and , with small oversampling () often sufficing. The practical impact is a scalable, robust preconditioning framework for GMRES on large, high-frequency Helmholtz problems, with coarse solvers that are simple, local, and decoupled from the subdomain details.

Abstract

In this work, we propose and analyze two two-level hybrid Schwarz preconditioners for solving the Helmholtz equation with high wave number in two and three dimensions. Both preconditioners are defined over a set of overlapping subdomains, with each preconditioner formed by a global coarse solver and one local solver on each subdomain. The global coarse solver is based on the localized orthogonal decomposition (LOD) technique, which was proposed in [30,31] originally for the discretization schemes for elliptic multiscale problems with heterogeneous and highly oscillating coefficients and Helmholtz problems with high wave number to eliminate the pollution effect. The local subproblems are Helmholtz problems in subdomains with homogeneous boundary conditions (the first preconditioner) or impedance boundary conditions (the second preconditioner). Both preconditioners are shown to be optimal under some reasonable conditions, that is, a uniform upper bound of the preconditioned operator norm and a uniform lower bound of the field of values are established in terms of all the key parameters, such as the fine mesh size, the coarse mesh size, the subdomain size and the wave numbers. It is the first time to show that the LOD solver can be a very effective coarse solver when it is used appropriately in the Schwarz method with multiple overlapping subdomains. Numerical experiments are presented to confirm the optimality and efficiency of the two proposed preconditioners.
Paper Structure (21 sections, 23 theorems, 179 equations, 5 tables)

This paper contains 21 sections, 23 theorems, 179 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite Element Method and Domain Decomposition
  4. Preliminary estimates
  5. Localized Orthogonal Decomposition (LOD) method
  6. LOD Coarse Spaces
  7. Well-posedness of coarse solvers
  8. Localized error estimates
  9. Two hybrid Schwarz preconditioners
  10. Dirichlet type preconditioner
  11. A priori estimates in the subdomains
  12. Optimality estimates of the first preconditioner $Q_{m}^{(1)}$
  13. Impedance type preconditioner
  14. A priori estimates in the subdomains
  15. Optimality estimates of the second preconditioner $Q_{m}^{(2)}$
  16. ...and 6 more sections

Key Result

Lemma 2.1

\newlabellem:multr0 If domain $D$ is of characteristic length $L$, it holds for any $v\in H^{1}(D)$ that grisvard_elliptic_1985 and it further holds for all $v\in H^1(D)$ vanishing on a subset of $\partial D$ with measure $L^{d-1}$ that where $C_{M}$ and $C_{F}$ depend only on $D$ and $C_{tr}=C_{M}^{1/2}(C_{F}(1+C_{F}))^{1/2}$.

Theorems & Definitions (42)

  • Lemma 2.1
  • Lemma 2.2
  • Proof 1
  • Lemma 2.3: graham_domain_2020, Lemma 3.3
  • Lemma 2.4
  • Lemma 2.5
  • Proof 2
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 32 more