pith. sign in

arxiv: 2606.04982 · v1 · pith:BCFU3KMAnew · submitted 2026-06-03 · 🧮 math.NA · cs.NA

Convergence of parallel overlapping domain decomposition methods with impedance boundary conditions for time-harmonic Maxwell equations in heterogeneous media

Luyu Cen , Shihua Gong , Euan A. Spence , Yue Yu This is my paper

Pith reviewed 2026-06-28 05:32 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords domain decompositionimpedance boundary conditionstime-harmonic Maxwell equationsconvergence analysisfinite element methodsheterogeneous mediaoverlapping subdomainsNedelec elements
0
0 comments X

The pith

Parallel overlapping domain decomposition for time-harmonic Maxwell equations converges when norms of impedance-to-impedance maps between subdomains are controlled.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes the convergence of parallel iterative domain decomposition methods using impedance boundary conditions for solving time-harmonic Maxwell equations in heterogeneous media. It shows that the method is well-posed and that the error propagation can be described using maps that capture how impedance conditions interact across subdomain boundaries. For decompositions into strips, explicit estimates on the convergence rate are derived in terms of these map norms. At the discrete level with finite elements, the method inherits the same convergence behavior provided the discrete maps approximate the continuous ones sufficiently well as the mesh is refined. This matters because such methods allow solving large-scale electromagnetic problems by breaking them into smaller subproblems that can be solved in parallel.

Core claim

The parallel iterative method is well-posed in an appropriate function space, and the error propagation operator is characterized through impedance-to-impedance maps that describe interactions between neighboring subdomains. For strip domain decompositions, explicit convergence estimates are derived in terms of the norms of the impedance-to-impedance maps. The finite-element counterpart based on Nédélec-element discretisations inherits the convergence behavior of the continuous method under the assumption that the discrete impedance-to-impedance maps approximate their continuous counterparts as the mesh is refined.

What carries the argument

impedance-to-impedance maps that describe interactions between neighboring subdomains and characterize the error propagation operator

If this is right

  • The parallel method converges for strip domain decompositions whenever the norms of the impedance-to-impedance maps are sufficiently small.
  • The discrete finite-element method with Nédélec elements inherits the same convergence estimates when the discrete maps approximate the continuous maps under mesh refinement.
  • Numerical experiments confirm the predicted convergence behavior for strip decompositions.
  • The same numerical framework applies to checkerboard decompositions even though the explicit estimates are not proved for that case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The impedance-to-impedance map characterization could be used to analyze convergence rates for other domain decomposition geometries or different wave equations.
  • Tuning the impedance parameters to reduce the norms of the maps might accelerate convergence in practice.
  • The robustness shown for heterogeneous media suggests the approach remains effective when material coefficients vary sharply across subdomains.

Load-bearing premise

The discrete impedance-to-impedance maps approximate their continuous counterparts as the mesh is refined.

What would settle it

A numerical computation in which the mesh is refined but the observed convergence rate of the discrete iterative method fails to match or approaches the rate predicted by the continuous impedance-to-impedance map analysis.

Figures

Figures reproduced from arXiv: 2606.04982 by Euan A. Spence, Luyu Cen, Shihua Gong, Yue Yu.

Figure 1
Figure 1. Figure 1: The field of values (FOV) of RAS-imp (with 2 subdomains) for different values of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Cross-sectional illustration of a strip decomposition (where Ω and each Ω [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Relative residual histories of the preconditioned Richardson iteration for a strip domain [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Relative residual histories of the preconditioned Richardson iteration for a strip domain [PITH_FULL_IMAGE:figures/full_fig_p031_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Canonical domain for the two-dimensional Maxwell equation. [PITH_FULL_IMAGE:figures/full_fig_p033_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Checkerboard domain decomposition into overlapping subdomains. [PITH_FULL_IMAGE:figures/full_fig_p034_6.png] view at source ↗
read the original abstract

This paper analyzes the convergence of parallel overlapping domain-decomposition methods with impedance boundary conditions for the time-harmonic Maxwell equations in heterogeneous media. We prove that the parallel iterative method is well-posed in an appropriate function space, and characterize the error propagation operator through impedance-to-impedance maps that describe interactions between neighboring subdomains. For strip domain decompositions, we derive explicit convergence estimates in terms of the norms of the impedance-to-impedance maps. At the discrete level, we develop the finite-element counterpart of these results based on N\'{e}d\'{e}lec-element discretisations. Under the assumption that the discrete impedance-to-impedance maps approximate their continuous counterparts as the mesh is refined, we show that the discrete method inherits the convergence behavior of the continuous method. We illustrate this theory with numerical experiments for strip domain decompositions, and also present numerical experiments for checkerboard domain decompositions that go beyond our theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper analyzes convergence of parallel overlapping domain decomposition methods with impedance boundary conditions for time-harmonic Maxwell equations in heterogeneous media. It proves well-posedness of the iterative method in an appropriate function space, characterizes the error propagation operator via impedance-to-impedance maps between neighboring subdomains, and derives explicit convergence estimates for strip decompositions in terms of the norms of these maps. At the discrete level, Nédélec finite-element discretizations are shown to inherit the continuous convergence behavior under the assumption that discrete impedance-to-impedance maps approximate their continuous counterparts as the mesh is refined. Numerical experiments illustrate the theory for strip decompositions and extend to checkerboard decompositions beyond the theory.

Significance. If the results hold, the continuous analysis supplies a useful characterization of error propagation through impedance-to-impedance maps together with explicit, strip-specific convergence estimates; this is a clear strength for the theoretical understanding of domain-decomposition solvers for Maxwell equations in heterogeneous media. The discrete extension, however, remains conditional on an unverified approximation property.

major comments (1)
  1. [Abstract and discrete analysis] Abstract and discrete finite-element section: the statement that the discrete method inherits the continuous convergence behavior rests on the assumption that discrete impedance-to-impedance maps approximate their continuous counterparts upon mesh refinement. No proof, error bound, or sufficient condition for this approximation is supplied, despite the presence of heterogeneous coefficients and the non-local character of the maps. This assumption is load-bearing for all discrete claims.
minor comments (1)
  1. [Numerical experiments] The numerical experiments for checkerboard decompositions are presented without accompanying theory; a short remark on observed convergence rates or possible extensions would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting both the strengths of the continuous analysis and the conditional nature of the discrete results. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and discrete analysis] Abstract and discrete finite-element section: the statement that the discrete method inherits the continuous convergence behavior rests on the assumption that discrete impedance-to-impedance maps approximate their continuous counterparts upon mesh refinement. No proof, error bound, or sufficient condition for this approximation is supplied, despite the presence of heterogeneous coefficients and the non-local character of the maps. This assumption is load-bearing for all discrete claims.

    Authors: We agree that the discrete inheritance result is conditional on the stated approximation property of the discrete impedance-to-impedance maps. The manuscript presents this explicitly as an assumption rather than a proven fact, precisely because establishing rigorous error bounds or sufficient conditions for these non-local maps in the presence of heterogeneous coefficients lies outside the scope of the present work. The continuous theory and the explicit strip-decomposition estimates are unconditional; the discrete section then shows that, whenever the approximation property holds, the discrete method inherits the same convergence behavior. Numerical experiments in the paper are consistent with this inheritance for the meshes considered. We therefore view the conditional discrete result as a useful bridge between the continuous analysis and practical computation, while acknowledging that a full proof of the approximation property would constitute a separate, technically demanding contribution. revision: no

Circularity Check

0 steps flagged

No circularity: continuous analysis self-contained; discrete part conditional on explicit external assumption

full rationale

The paper proves well-posedness of the parallel iterative method and characterizes the error propagation operator via impedance-to-impedance maps for the continuous problem, then derives explicit estimates for strip decompositions directly from norms of those maps. The discrete Nédélec FE version is stated to inherit the same behavior only under the separate assumption that discrete maps approximate continuous ones under mesh refinement; this assumption is not derived from the result itself, fitted from data, or justified by self-citation chains. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides limited information; relies on standard mathematical assumptions for PDE well-posedness and FEM approximation without specifying free parameters or invented entities.

axioms (1)
  • domain assumption The time-harmonic Maxwell equations are well-posed in appropriate function spaces for heterogeneous media
    Invoked to establish the continuous problem setup and well-posedness of the iterative method.

pith-pipeline@v0.9.1-grok · 5703 in / 1298 out tokens · 37190 ms · 2026-06-28T05:32:14.942756+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

43 extracted references · 4 canonical work pages

  1. [1]

    I. M. Babuska and S. A. Sauter. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?SIAM Review, 42(3):451–484, 2000

    2000

  2. [2]

    J. M. Ball, Y. Capdeboscq, and B. Tsering-Xiao. On uniqueness for time harmonic anisotropic Maxwell’s equations with piecewise regular coefficients.Mathematical Models and Methods in Applied Sciences, 22(11):1250036, 11, 2012

    2012

  3. [3]

    Beckermann, S

    B. Beckermann, S. A. Goreinov, and E. E. Tyrtyshnikov. Some remarks on the Elman estimate for GMRES. SIAM Journal on Matrix Analysis and Applications, 27(3):772–778, 2006

    2006

  4. [4]

    Bonazzoli, P

    M. Bonazzoli, P. Ciarlet, A. Modave, and A. Rappaport. Analysis of a two-level domain decomposition preconditioner for the time-harmonic Maxwell equations in anisotropic media.hal-05572507, 2026

    2026

  5. [5]

    Bonazzoli, V

    M. Bonazzoli, V. Dolean, I. G. Graham, E. A. Spence, and P.-H. Tournier. Domain decomposition pre- conditioning for the high-frequency time-harmonic Maxwell equations with absorption.Mathematics of Computation, 88(320):2559–2604, 2019

    2019

  6. [6]

    Bootland, V

    N. Bootland, V. Dolean, P. Jolivet, and P.-H. Tournier. A comparison of coarse spaces for Helmholtz problems in the high frequency regime.Computers&Mathematics with Applications, 98:239–253, 2021

    2021

  7. [7]

    S. N. Chandler-Wilde and E. A. Spence. Coercive second-kind boundary integral equations for the Laplace Dirichlet problem on Lipschitz domains.Numerische Mathematik, 156(4):1325–1384, 2024

    2024

  8. [8]

    Chaumont-Frelet, J

    T. Chaumont-Frelet, J. Galkowski, and E. A. Spence. Sharp error bounds for edge-element discretisations of the high-frequency Maxwell equations.Math. Comp., DOI 10.1090/mcom/4198, 2026

    work page doi:10.1090/mcom/4198 2026

  9. [9]

    Chaumont-Frelet, A

    T. Chaumont-Frelet, A. Moiola, and E. A. Spence. Explicit bounds for the high-frequency time-harmonic Maxwell equations in heterogeneous media.Journal de Math´ ematiques Pures et Appliqu´ ees, 179:183–218, 2023

    2023

  10. [10]

    Costabel

    M. Costabel. A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains.Mathe- matical Methods in the Applied Sciences, 12(4):365–368, 1990

    1990

  11. [11]

    Costabel and M

    M. Costabel and M. Dauge. Singularities of electromagnetic fields in polyhedral domains.Archive for Rational Mechanics and Analysis, 151(3):221–276, 2000

    2000

  12. [12]

    Costabel, M

    M. Costabel, M. Dauge, and S. Nicaise. Singularities of Maxwell interface problems.ESAIM: Mathematical Modelling and Numerical Analysis, 33(3):627–649, 1999

    1999

  13. [13]

    Costabel, M

    M. Costabel, M. Dauge, and S. Nicaise. Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth domains. 2010.https://hal.archives-ouvertes.fr/file/index/docid/453934/ filename/CoDaNi_Analytic_Part_I.pdf

    2010

  14. [14]

    S. C. Eisenstat, H. C. Elman, and M. H. Schultz. Variational iterative methods for nonsymmetric systems of linear equations.SIAM Journal on Numerical Analysis, pages 345–357, 1983

    1983

  15. [15]

    H. C. Elman.Iterative Methods for Sparse Nonsymmetric Systems of Linear Equations. PhD thesis, Yale University, 1982

    1982

  16. [16]

    L. C. Evans and R. F. Gariepy.Measure theory and fine properties of functions. Chapman and Hall/CRC, 1992

    1992

  17. [17]

    Feng and H

    X. Feng and H. Wu. An absolutely stable discontinuous Galerkin method for the indefinite time-harmonic Maxwell equations with large wave number.SIAM Journal on Numerical Analysis, 52(5):2356–2380, 2014

    2014

  18. [18]

    Galkowski, S

    J. Galkowski, S. Gong, I. G. Graham, David Lafontaine, and E. A. Spence. Convergence of overlapping do- main decomposition methods with PML transmission conditions applied to nontrapping Helmholtz problems. arXiv preprint arXiv:2404.02156, 2024

    work page arXiv 2024

  19. [19]

    Girault and P.-A

    V. Girault and P.-A. Raviart.Finite element methods for Navier-Stokes equations: theory and algorithms. Springer–Verlag, 1986

    1986

  20. [20]

    S. Gong, M. J. Gander, I. G. Graham, David Lafontaine, and E. A. Spence. Convergence of parallel overlap- ping domain decomposition methods for the Helmholtz equation.Numerische Mathematik, 152(2):259–306, 2022

    2022

  21. [21]

    S. Gong, I. G. Graham, and E. A. Spence. Convergence of restricted additive Schwarz with impedance transmission conditions for discretised Helmholtz problems.Mathematics of Computation, 92(339):175–215, 2023

    2023

  22. [22]

    Gopalakrishnan and J

    J. Gopalakrishnan and J. Pasciak. Overlapping Schwarz preconditioners for indefinite time harmonic Maxwell equations.Mathematics of computation, 72(241):1–15, 2003

    2003

  23. [23]

    Grisvard.Elliptic problems in nonsmooth domains

    P. Grisvard.Elliptic problems in nonsmooth domains. Pitman, Boston, 1985

    1985

  24. [24]

    Hiptmair

    R. Hiptmair. Finite elements in computational electromagnetism.Acta Numerica, 11:237–339, 2002. 37

    2002

  25. [25]

    Hiptmair and J

    R. Hiptmair and J. Xu. Nodal auxiliary space preconditioning in H (curl) and H (div) spaces.SIAM Journal on Numerical Analysis, 45(6):2483–2509, 2007

    2007

  26. [26]

    Jerison and C

    D. Jerison and C. E. Kenig. The inhomogeneous Dirichlet problem in Lipschitz domains.Journal of Func- tional Analysis, 130(1):161–219, 1995

    1995

  27. [27]

    Lafontaine and E

    D. Lafontaine and E. A. Spence. Sharp bounds on Helmholtz impedance-to-impedance maps and application to overlapping domain decomposition.Pure and Applied Analysis, 5(4):927–972, 2023

    2023

  28. [28]

    Li and Q

    Z. Li and Q. Hu. A hybrid two-level weighted Schwarz method for time-harmonic Maxwell equations.Journal of Computational and Applied Mathematics, 475(117015), 2025

    2025

  29. [29]

    H. Liu, L. Rondi, and J. Xiao. Mosco convergence forH(curl) spaces, higher integrability for Maxwell’s equations, and stability in direct and inverse EM scattering problems.Journal of the European Mathematical Society, 21(10):2945–2993, 2019

    2019

  30. [30]

    P. Lu, H. Wu, and X. Xu. Continuous interior penalty finite element methods for the time-harmonic Maxwell equation with high wave number.Advances in Computational Mathematics, 45(5):3265–3291, 2019

    2019

  31. [31]

    Lu and H

    S. Lu and H. Wu. Preasymptotic error estimates of higher-order EEM for the time-harmonic Maxwell equations with large wave number.arXiv preprint arXiv:2504.19481, 2025

    work page arXiv 2025

  32. [32]

    Lu and H

    S. Lu and H. Wu. Preasymptotic error estimates of linear EEM and CIP-EEM for the time-harmonic Maxwell equations with large wave number.SIAM Journal on Numerical Analysis, 64(2):326–349, 2026

    2026

  33. [33]

    W. C. H. McLean.Strongly elliptic systems and boundary integral equations. Cambridge University Press, 2000

    2000

  34. [34]

    J. M. Melenk and S. A. Sauter. Wavenumber-explicithp-FEM analysis for Maxwell’s equations with impedance boundary conditions.Foundations of Computational Mathematics, 24(6):1871–1939, 2024

    1939

  35. [35]

    J. M. Melenk and D. W¨ org¨ otter. Wavenumber-explicithp-FEM analysis of Maxwell’s equations with impedance boundary conditions in piecewise smooth media.arXiv preprint arXiv:2603.17467, 2026

    work page arXiv 2026

  36. [36]

    L. Miller. Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary.Journal de math´ ematiques pures et appliqu´ ees, 79(3):227–269, 2000

    2000

  37. [37]

    Moiola.Trefftz-discontinuous Galerkin methods for time-harmonic wave problems

    A. Moiola.Trefftz-discontinuous Galerkin methods for time-harmonic wave problems. PhD thesis, ETH Zurich, 2011

    2011

  38. [38]

    Monk.Finite element methods for Maxwell’s equations

    P. Monk.Finite element methods for Maxwell’s equations. Oxford University Press, 2003

    2003

  39. [39]

    Neˇ cas.Les m´ ethodes directes en th´ eorie des ´ equations elliptiques

    J. Neˇ cas.Les m´ ethodes directes en th´ eorie des ´ equations elliptiques. Masson, 1967

    1967

  40. [40]

    Nicaise and J

    S. Nicaise and J. Tomezyk. The time-harmonic Maxwell equations with impedance boundary conditions in polyhedral domains. InMaxwell’s Equations: Analysis and Numerics, volume 24 ofRadon Series on Computational and Applied Mathematics, pages 285–340. 2019

    2019

  41. [41]

    Nicaise and J

    S. Nicaise and J. Tomezyk. Convergence analysis of a hp-finite element approximation of the time-harmonic Maxwell equations with impedance boundary conditions in domains with an analytic boundary.Numerical Methods for Partial Differential Equations, 36(6):1868–1903, 2020

    1903

  42. [42]

    E. A. Spence. Preconditioning FEM discretisations of the high-frequency Helmholtz and Maxwell equations by either perturbing the coefficients or adding absorption.Computers&Mathematics with Applications, 210:186–199, 2026

    2026

  43. [43]

    Zworski.Semiclassical analysis, volume 138

    M. Zworski.Semiclassical analysis, volume 138. American Mathematical Soc., 2012. 38

    2012