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A novel coarse space applying to the weighted Schwarz method for Helmholtz equations

Qiya Hu, Ziyi Li

TL;DR

The paper addresses efficiently solving high-frequency Helmholtz equations using a two-level weighted restricted additive Schwarz (WASI) framework. It introduces an adaptive coarse space built from local Helmholtz-harmonic eigenfunctions defined on impedance-boundary subspaces and couples it with a WASI one-level preconditioner to form a two-level hybrid method, proving uniform convergence with respect to mesh size $h$, subdomain size $d$, and wave number $\\kappa$ under specific parameter regimes. An economical coarse-space variant is proposed to avoid eigenproblems while preserving accuracy, and the authors provide comprehensive numerical experiments that corroborate the theory and demonstrate practical gains over existing coarse-space approaches. The results offer a scalable, robust preconditioning strategy for indefinite Helmholtz systems in 2D, with clear guidance on parameter choices and coarse-space design for different absorption regimes.

Abstract

In this paper we are concerned with restricted additive Schwarz with local impedance transformation conditions for a family of Helmholtz problems in two dimensions. These problems are discretized by the finite element method with conforming nodal finite elements. We design and analyze a new adaptive coarse space for this kind of restricted additive Schwarz method. This coarse space is spanned by some eigenvalue functions of local generalized eigenvalue problems, which are defined by weighted positive semi-definite bilinear forms on subspaces consisting of local discrete Helmholtz-harmonic functions from impedance boundary data. We proved that a two-level hybrid Schwarz preconditioner with the proposed coarse space possesses uniformly convergence independent of the mesh size, the subdomain size and the wave numbers under suitable assumptions. We also introduce an economic coarse space to avoid solving generalized eigenvalue problems. Numerical experiments confirm the theoretical results.

A novel coarse space applying to the weighted Schwarz method for Helmholtz equations

TL;DR

The paper addresses efficiently solving high-frequency Helmholtz equations using a two-level weighted restricted additive Schwarz (WASI) framework. It introduces an adaptive coarse space built from local Helmholtz-harmonic eigenfunctions defined on impedance-boundary subspaces and couples it with a WASI one-level preconditioner to form a two-level hybrid method, proving uniform convergence with respect to mesh size , subdomain size , and wave number under specific parameter regimes. An economical coarse-space variant is proposed to avoid eigenproblems while preserving accuracy, and the authors provide comprehensive numerical experiments that corroborate the theory and demonstrate practical gains over existing coarse-space approaches. The results offer a scalable, robust preconditioning strategy for indefinite Helmholtz systems in 2D, with clear guidance on parameter choices and coarse-space design for different absorption regimes.

Abstract

In this paper we are concerned with restricted additive Schwarz with local impedance transformation conditions for a family of Helmholtz problems in two dimensions. These problems are discretized by the finite element method with conforming nodal finite elements. We design and analyze a new adaptive coarse space for this kind of restricted additive Schwarz method. This coarse space is spanned by some eigenvalue functions of local generalized eigenvalue problems, which are defined by weighted positive semi-definite bilinear forms on subspaces consisting of local discrete Helmholtz-harmonic functions from impedance boundary data. We proved that a two-level hybrid Schwarz preconditioner with the proposed coarse space possesses uniformly convergence independent of the mesh size, the subdomain size and the wave numbers under suitable assumptions. We also introduce an economic coarse space to avoid solving generalized eigenvalue problems. Numerical experiments confirm the theoretical results.
Paper Structure (22 sections, 14 theorems, 125 equations, 5 tables)

This paper contains 22 sections, 14 theorems, 125 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Variational problem and its discretization
  4. Notations
  5. Some basic auxiliary results
  6. A domain decomposition preconditioner
  7. One level settings
  8. A coarse space
  9. Projection operators
  10. Some auxiliary results
  11. Convergence analysis for the case with a slightly large $\epsilon$
  12. Convergence analysis for the case with a small $\epsilon$
  13. Some other coarse spaces
  14. Several existing coarse spaces
  15. Coarse space based on the coarse grid
  16. ...and 7 more sections

Key Result

Lemma 2.1

(Multiplicative trace inequality) Let $G$ be a Lipschitz domain with characteristic length scale $L$, then for every $v\in H^1(G)$ there holds

Theorems & Definitions (31)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Remark 2.1
  • Lemma 3.1
  • proof
  • proof
  • Lemma 4.1
  • ...and 21 more