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A hybrid two-level weighted Schwartz method for time-harmonic Maxwell equations

Ziyi Li, Qiya Hu

TL;DR

The paper tackles robust preconditioning for large, indefinite linear systems from time-harmonic Maxwell equations with absorption, discretized by Nédélec finite elements. It develops a two-level hybrid Schwarz preconditioner with impedance subproblems, where an adaptive coarse space is constructed from local Maxwell-harmonic eigenfunctions controlled by a tolerance $\rho$, and it proves discrete stability and inf-sup bounds that yield GMRES convergence independent of $h$, $H$, and $\kappa$ for suitably small $\rho$. An economical variant replaces expensive eigenproblems with a boundary-based spherical-harmonic construction to preserve efficiency. Numerical experiments on constant-wavenumber and layered-media models confirm the theoretical robustness and demonstrate practical performance gains, including reduced coarse-space cost without sacrificing convergence.

Abstract

This paper concerns the preconditioning technique for discrete systems arising from time-harmonic Maxwell equations with absorptions, where the discrete systems are generated by Nédélec finite element methods of fixed order on meshes with suitable size. This kind of preconditioner is defined as a two-level hybrid form, which falls into the class of ``unsymmetrically weighted'' Schwarz method based on the overlapping domain decomposition with impedance boundary subproblems. The coarse space in this preconditioner is constructed by specific eigenfunctions solving a series of generalized eigenvalue problems in the local discrete Maxwell-harmonic spaces according to a user-defined tolerance $ρ$. We establish a stability result for the considered discrete variational problem. Using this discrete stability, we prove that the two-level hybrid Schwarz preconditioner is robust in the sense that the convergence rate of GMRES is independent of the mesh size, the subdomain size and the wave-number when $ρ$ is chosen appropriately. We also define an economical variant that avoids solving generalized eigenvalue problems. Numerical experiments confirm the theoretical results and illustrate the efficiency of the preconditioners.

A hybrid two-level weighted Schwartz method for time-harmonic Maxwell equations

TL;DR

The paper tackles robust preconditioning for large, indefinite linear systems from time-harmonic Maxwell equations with absorption, discretized by Nédélec finite elements. It develops a two-level hybrid Schwarz preconditioner with impedance subproblems, where an adaptive coarse space is constructed from local Maxwell-harmonic eigenfunctions controlled by a tolerance , and it proves discrete stability and inf-sup bounds that yield GMRES convergence independent of , , and for suitably small . An economical variant replaces expensive eigenproblems with a boundary-based spherical-harmonic construction to preserve efficiency. Numerical experiments on constant-wavenumber and layered-media models confirm the theoretical robustness and demonstrate practical performance gains, including reduced coarse-space cost without sacrificing convergence.

Abstract

This paper concerns the preconditioning technique for discrete systems arising from time-harmonic Maxwell equations with absorptions, where the discrete systems are generated by Nédélec finite element methods of fixed order on meshes with suitable size. This kind of preconditioner is defined as a two-level hybrid form, which falls into the class of ``unsymmetrically weighted'' Schwarz method based on the overlapping domain decomposition with impedance boundary subproblems. The coarse space in this preconditioner is constructed by specific eigenfunctions solving a series of generalized eigenvalue problems in the local discrete Maxwell-harmonic spaces according to a user-defined tolerance . We establish a stability result for the considered discrete variational problem. Using this discrete stability, we prove that the two-level hybrid Schwarz preconditioner is robust in the sense that the convergence rate of GMRES is independent of the mesh size, the subdomain size and the wave-number when is chosen appropriately. We also define an economical variant that avoids solving generalized eigenvalue problems. Numerical experiments confirm the theoretical results and illustrate the efficiency of the preconditioners.

Paper Structure

This paper contains 18 sections, 23 theorems, 164 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notations
  4. Variational problem and its discretization
  5. Basic properties of the sesquilinear forms
  6. Stability results for problems with absorption
  7. A hybrid Schwarz preconditioner
  8. One level preconditioner
  9. An adaptive coarse space
  10. The main convergence results
  11. Some auxiliary results
  12. Convergence analysis of the preconditioner
  13. An economical preconditioner
  14. Numerical experiments
  15. Model with constant wavenumber
  16. ...and 3 more sections

Key Result

Lemma 2.1

(1) Let $0\leq\epsilon\leq\kappa^2$, then the sesquilinear forms $a_\epsilon(\cdot,\cdot)$ and $\bar{a}_\epsilon(\cdot,\cdot)$ are continuous: for all ${\boldsymbol{\mathbf{v}}},{\boldsymbol{\mathbf{w}}}\in H_\mathrm{imp}({\boldsymbol{\mathbf{curl}}};\Omega)$. (2) Let $\epsilon>0$. The sesquilinear forms $a_{\epsilon}(\cdot,\cdot)$ and $\bar{a}_\epsilon(\cdot,\cdot)$ are coercive: for all ${\bol

Theorems & Definitions (40)

  • Lemma 2.1: Continuity and coercivity, bonazzoli2019domain
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 30 more