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arxiv: 2605.15636 · v1 · pith:IEYVL62Rnew · submitted 2026-05-15 · 🧮 math.NA · cs.NA

A Tearing and Interconnecting Formulation for Magneto-Quasi-Statics

Clemens Pechstein This is my paper

Pith reviewed 2026-05-19 22:30 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords tearing and interconnectingdomain decompositionmagneto-quasi-staticseddy current modelvector potentialgradient splittinginterface kernelsH(div) space
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The pith

A space splitting into gradient fields and a complementary part makes subdomain operators invertible in a tearing-and-interconnecting formulation for the eddy current model.

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

The paper develops a non-overlapping domain decomposition method for magneto-quasi-static problems that involve one conducting subdomain and one insulating subdomain. Standard tree-cotree splittings leave many kernel functions tied to the interface, so the authors replace that approach with an explicit splitting of the function space into gradient fields and a complementary space. Under a mild condition on this splitting, the non-conducting subdomain requires no gradient component at all, which removes any need to couple gradient degrees of freedom across the interface. Both resulting subdomain operators become invertible, and the magnetic field recovered from the discontinuous vector potential remains globally divergence-conforming. This structure directly addresses the kernel and interface issues that otherwise block stable domain-decomposition solvers for the eddy-current equations.

Core claim

Under a mild condition on the splitting of the space into gradient fields and a complementary space, one does not need any gradient part in the non-conducting domain and therefore no coupling of any gradient components between the two subdomains, both subdomain operators are invertible, and although the magnetic vector potential is discontinuous across the subdomain interface, the corresponding magnetic field is globally in H(div).

What carries the argument

The splitting of the relevant function space into gradient fields and a complementary space, which removes interface-associated kernels and enables the three listed properties without additional gradient coupling.

If this is right

  • Subdomain problems can be solved independently because no gradient components need to be matched across the interface.
  • The global magnetic field satisfies the required regularity even though the vector potential itself jumps at the subdomain boundary.
  • Kernel issues that survive tree-cotree splittings are removed once the complementary-space condition holds.
  • The formulation supports non-overlapping domain decomposition without extra Lagrange multipliers for gradient continuity.

Where Pith is reading between the lines

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

  • The mild condition may be satisfied automatically by standard choices of tree-cotree bases on practical unstructured meshes.
  • The same splitting idea could be applied when more than two subdomains are present.
  • The H(div) continuity result may simplify post-processing steps that compute derived quantities such as forces or losses.

Load-bearing premise

The chosen splitting of the space into gradient fields and a complementary space satisfies the mild condition that eliminates the interface kernels.

What would settle it

For a concrete mesh and a splitting that meets the mild condition, assemble the two subdomain operators and check whether either has a nontrivial kernel or whether the reconstructed magnetic field fails to lie in the global H(div) space.

read the original abstract

This note deals with a tearing and interconnecting (special non-overlapping domain decomposition) formulation for magneto-quasi-statics (also known as the eddy current model). Only two subdomains are considered, one conducting and one insulating. Using a straightforward tree-cotree splitting, one can get rid of some kernel components in the non-conducting region, but due to the coupling across the interface, a lot of kernel functions remain that are associated with the interface. The formulation presented here overcomes this problem by using a space splitting into gradient fields and a complementary space. Under a mild condition on that splitting, it is shown that (i) one does not need any gradient part in the non-conducting domain, and therefore no coupling of any gradient components between the two subdomains, (ii) both subdomain operators are invertible, and (iii) although the magnetic vector potential is discontinuous across the subdomain interface, the corresponding magnetic field is globally in H(div).

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

2 major / 2 minor

Summary. The paper develops a tearing and interconnecting (non-overlapping domain decomposition) formulation for the magneto-quasi-static (eddy-current) model on two subdomains, one conducting and one insulating. It employs a tree-cotree splitting followed by a further decomposition of the space into gradient fields and a complementary space. Under a mild condition on this splitting, the authors claim that (i) the non-conducting subdomain requires no gradient component and thus no gradient coupling across the interface, (ii) both subdomain operators become invertible, and (iii) the magnetic vector potential may jump across the interface while the magnetic field remains globally in H(div).

Significance. If the central claims hold, the formulation removes a known source of interface kernels in domain-decomposition methods for eddy-current problems and guarantees a physically consistent global magnetic field. The approach could simplify implementation and improve conditioning in large-scale simulations, provided the mild condition is readily satisfiable on standard meshes.

major comments (2)
  1. [Section introducing the space splitting (near the statement of the mild condition)] The mild condition on the gradient/complementary splitting is invoked to prove claims (i)–(iii) and to remove interface kernels, yet its precise statement, its relation to the tree-cotree decomposition, and its verification for typical finite-element spaces are not made fully explicit. This condition is load-bearing for all three main results.
  2. [The subsection establishing invertibility of the subdomain operators] The proof that both subdomain operators are invertible (claim (ii)) relies on the chosen splitting to eliminate the kernel; the manuscript should supply a self-contained argument showing that the complementary space is chosen so that the only harmonic fields remaining are those already controlled by the conducting subdomain.
minor comments (2)
  1. [Notation and preliminaries] Notation for the interface trace operators and the precise definition of the complementary space should be introduced earlier and used consistently throughout the proofs.
  2. [Numerical results or concluding remarks] A brief numerical example or reference to a standard test mesh would help the reader confirm that the mild condition can be met in practice.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and will revise the manuscript accordingly to improve explicitness and completeness.

read point-by-point responses

  1. Referee: [Section introducing the space splitting (near the statement of the mild condition)] The mild condition on the gradient/complementary splitting is invoked to prove claims (i)–(iii) and to remove interface kernels, yet its precise statement, its relation to the tree-cotree decomposition, and its verification for typical finite-element spaces are not made fully explicit. This condition is load-bearing for all three main results.

    Authors: We agree that a more explicit statement of the mild condition, together with its precise relation to the tree-cotree decomposition and verification on standard meshes, would strengthen the presentation. In the revised manuscript we will add a dedicated paragraph immediately following the definition of the splitting that states the condition in mathematical terms, explains how it is satisfied by a standard tree-cotree construction on tetrahedral meshes, and provides a short verification argument showing that the complementary space can always be chosen to meet the condition for any conforming finite-element space on a simply-connected insulating subdomain. revision: yes

  2. Referee: [The subsection establishing invertibility of the subdomain operators] The proof that both subdomain operators are invertible (claim (ii)) relies on the chosen splitting to eliminate the kernel; the manuscript should supply a self-contained argument showing that the complementary space is chosen so that the only harmonic fields remaining are those already controlled by the conducting subdomain.

    Authors: The referee is right that the current argument for invertibility would be clearer if presented in a fully self-contained manner. We will expand the subsection to include a direct proof that, once the mild condition is satisfied, any harmonic field belonging to the complementary space in the insulating subdomain must be identically zero. The argument proceeds by showing that such a field would extend to a globally harmonic field that is orthogonal to the conducting subdomain’s tree-cotree basis, contradicting the interface transmission conditions and the fact that all harmonic degrees of freedom are already fixed by the conducting region. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a mathematical tearing-and-interconnecting formulation for the eddy-current model, relying on a tree-cotree splitting followed by a gradient/complementary space decomposition. Under an explicitly stated mild condition on that splitting, it directly proves the three listed properties (no gradient coupling needed, subdomain operators invertible, global H(div) regularity) via standard functional-analysis arguments on the interface kernels. No equations reduce to self-definition, no parameters are fitted and then relabeled as predictions, and no load-bearing claims rest on self-citations; the derivation is self-contained within the given assumptions and the cited external literature on domain decomposition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard background results from finite element theory for Maxwell equations plus one key domain-specific assumption.

axioms (1)
  • domain assumption Mild condition on the splitting into gradient fields and a complementary space
    Invoked to establish that no gradient part is needed in the non-conducting domain and that subdomain operators are invertible.

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Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [1]

    Albanese and G

    R. Albanese and G. Rubinacci. Solution of three dimensional eddy current problems by integral and differential methods.IEEE Trans. Magn., vol. 24(1):98–101, 1988

    work page 1988

  2. [2]

    Alonso Rodr´ ıguez and A

    A. Alonso Rodr´ ıguez and A. Valli.Eddy Current Approximation of Maxwell Equa- tions. Springer, Milan, 2010

    work page 2010

  3. [3]

    Buffa, M

    A. Buffa, M. Costabel, and D. Sheen. On traces for H(curl, Ω) in Lipschitz domains. J. Math. Anal. Appl., 276(2):845–867, 2002

    work page 2002

  4. [4]

    C. R. Dohrmann. A preconditioner for substructuring based on constrained energy minimization.SIAM J. Sci. Comput., 25(1):246–258, 2003

    work page 2003

  5. [5]

    C. R. Dohrmann and O. B. Widlund. A BDDC algorithm with deluxe scaling for three-dimensional H(curl) problems.Comm. Pure Appl. Math., 69(4):745–770, 2016

    work page 2016

  6. [6]

    Eller, S

    M. Eller, S. Reitzinger, S. Sch¨ ops, and S. Zaglmayr. A symmetric low-frequency stable broadband Maxwell formulation for industrial applications.J. Sci. Comput., 39(4):B703–B731, 2017

    work page 2017

  7. [7]

    Farhat, M

    C. Farhat, M. Lesoinne, P. Le Tallec, K. Pierson, and D. Rixen. FETI-DP: a dual- primal unified FETI method I: A faster alternative to the two-level FETI method. Int. J. Numer. Meth. Engng., 50(7):1523–1544, 2001

    work page 2001

  8. [8]

    Farhat, A

    C. Farhat, A. Macedo, M. Lesoinne, F. Roux, F. Magoul` es, and A. de La Bourdon- naie. Two-level domain decomposition methods with Lagrange multipliers for the fast iterative solution of acoustic scattering problems.Comput. Methods Appl. Mech. Engrg., 184:213–239, 2000

    work page 2000

  9. [9]

    Farhat and F

    C. Farhat and F. Roux. A method of finite element tearing and interconnecting and its parallel solution algorithm.Int. J. Numer. Meth. Engng., 32(6):1205–1227, 1991

    work page 1991

  10. [10]

    Farhat and F

    C. Farhat and F. Roux. Implicit parallel processing in structural mechanics. In T. Oden, editor,Computational Mechanics Advances, volume II, pages 1–124. North- Holland, Amsterdam, 1994. 13

    work page 1994

  11. [11]

    Hiptmair, F

    R. Hiptmair, F. Kr¨ amer, and J. Ostrowski. A robust Maxwell formulation for all frequencies.IEEE Trans. Magn., 44:682–685, 2008

    work page 2008

  12. [12]

    M. T. Jochum.Niederfrequenzstabile Potenzialformulierungen zur Finite-Elemente- Simulation elektromagnetischer Felder im Frequenzbereich. PhD thesis, Universit¨ at des Saarlandes, Saarbr¨ ucken, Germany, 2013

    work page 2013

  13. [13]

    Kapidani, M

    B. Kapidani, M. Merkel, S. Sch¨ ops, and R. V´ azquez. Tree-cotree decomposition of isogeometric mortared spaces in H(curl) on multi-patch domains.Comput. Methods Appl. Mech. Engrg., 395, 2022. Article 114949

    work page 2022

  14. [14]

    S. K. Kleiss, C. Pechstein, B. J¨ uttler, and S. Tomar. IETI – isogeometric tearing and interconnecting.Comput. Methods Appl. Mech. Engrg., 247-248(1), 2012

    work page 2012

  15. [15]

    Langer and O

    U. Langer and O. Steinbach. Boundary element tearing and interconnecting methods. Computing, 71:205–228, 2003

    work page 2003

  16. [16]

    Mally, B

    M. Mally, B. Kapidani, M. Merkel, S. Sch¨ ops, and R. V´ azquez. Tree–cotree-based tearing and interconnecting for 3D magnetostatics: A dual–primal approach.Com- put. Methods Appl. Mech. Engrg., 437(15), 2025

    work page 2025

  17. [17]

    Mandel and M

    J. Mandel and M. Brezina. Balancing domain decomposition for problems with large jumps in coefficients.Math. Comp., 65:1387–1401, 1996

    work page 1996

  18. [18]

    Mandel, C

    J. Mandel, C. R. Dohrmann, and R. Tezaur. An algebraic theory for primal and dual substructuring methods by constraints.Appl. Numer. Math., 54(2):167–193, 2005

    work page 2005

  19. [19]

    J. B. Manges and Z. J. Cendes. A generalized tree-cotree gauge for magnetic field computation.IEEE Trans. Magn., 31(3):1342–1347, 1995

    work page 1995

  20. [20]

    McLean.Strongly elliptic systems and boundary integral equations

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

    work page 2000

  21. [21]

    Monk.Finite Element Methods for Maxwell’s Equations

    P. Monk.Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York, 2003

    work page 2003

  22. [22]

    Of.BETI-Gebietszerlegungsmethoden mit schnellen Randelementverfahren und Anwendungen

    G. Of.BETI-Gebietszerlegungsmethoden mit schnellen Randelementverfahren und Anwendungen. PhD thesis, Universit¨ at Stuttgart, Germany, 2006

    work page 2006

  23. [23]

    G. N. Paraschos.Robust and scalable domain decomposition methods for electromag- netic computations. PhD thesis, University of Massachusetts Amherst, September 2012

    work page 2012

  24. [24]

    Pechstein.Finite and Boundary Element Tearing and Interconnecting Methods for Multiscale Problems, volume 90 ofLecture Notes in Computational Science and Engineering

    C. Pechstein.Finite and Boundary Element Tearing and Interconnecting Methods for Multiscale Problems, volume 90 ofLecture Notes in Computational Science and Engineering. Springer, Berlin, 2013

    work page 2013

  25. [25]

    Pechstein

    C. Pechstein. A unified theory of non-overlapping Robin-Schwarz methods: contin- uous and discrete, including cross points.J. Sci. Comput., 96(2), 2023

    work page 2023

  26. [26]

    Pechstein and C

    C. Pechstein and C. R. Dohrmann. A unified framework for adaptive BDDC.Elec- tron. Transact. Numer. Anal., 46:273–336, 2017. (electronic). 14

    work page 2017

  27. [27]

    Rapetti, A

    F. Rapetti, A. Alonso Rodr´ ıguez, and E. De Los Santos. On the tree gauge in magnetostatics.Multidisciplinary Scientific J., 5(1):52–63, 2022

    work page 2022

  28. [28]

    Schneckenleitner and S

    R. Schneckenleitner and S. Takacs. Condition number bounds for IETI-DP methods that are explicit inpandh.Math. Models Methods Appl. Sci., 30(11):2067–2103, 2020

    work page 2067

  29. [29]

    A. Toselli. Dual-primal FETI algorithms for edge finite-element approximations in 3D.IMA J. Numer. Anal., 26(1):96–130, 2006

    work page 2006

  30. [30]

    Toselli and O

    A. Toselli and O. B. Widlund.Domain Decomposition Methods – Algorithms and The- ory, volume 34 ofSpringer Series in Computational Mathematics. Springer, Berlin, 2005

    work page 2005

  31. [31]

    M. N. Vouvakis.A non-conformal domain decomposition method for solving large electromagnetic wave problems. PhD thesis, The Ohio State University, Columbus, Ohio, 2005

    work page 2005

  32. [32]

    Windisch.Boundary Element Tearing and Interconnecting Methods for Acoustic and Electromagnetic Scattering

    M. Windisch.Boundary Element Tearing and Interconnecting Methods for Acoustic and Electromagnetic Scattering. PhD thesis, Graz University of Technology, Austria, December 2011. 15

    work page 2011