pith. sign in

arxiv: 2606.24517 · v1 · pith:DDLDYCSNnew · submitted 2026-06-23 · 🧮 math.NA · cs.NA

Analysis and approximation of a two-dimensional induction heating problem

Gabriel R. Barrenechea , Katherine MacKenzie , Abner Salgado This is my paper

Pith reviewed 2026-06-25 23:14 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords induction heatingfinite element approximationexistence of solutionsweak formulationbound-preserving methodGalerkin methodnumerical analysis
0
0 comments X

The pith

Finite element convergence establishes existence of solutions to a two-dimensional induction heating problem

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

The paper examines the existence and finite element approximation of solutions to a steady-state two-dimensional induction heating problem, where the forcing term appears only integrable at first glance. A priori regularity results for the governing PDEs are used to justify the natural weak formulation. Convergence of the standard Galerkin method is proven in convex domains under appropriate mesh conditions, while the bound-preserving method extends this to more general domains and meshes. Because these convergence results hold without assuming extra regularity on the solutions, they simultaneously establish existence. Numerical tests illustrate the theory and the practical gains from the bound-preserving technique.

Core claim

By justifying the weak formulation through a priori regularity estimates and then proving convergence of finite element approximations without any regularity assumptions on the solutions, the analyses demonstrate both the existence of solutions to the induction heating problem and the reliability of the numerical methods.

What carries the argument

The a priori regularity results that justify the weak formulation, combined with convergence analysis of the Galerkin finite element method and the bound-preserving method applied to the heat equation.

If this is right

  • Existence of solutions is proven for the induction heating problem even with merely integrable data.
  • Standard finite element methods converge for the problem in convex domains with suitable meshes.
  • The bound-preserving method converges under weaker assumptions on the domain and mesh.
  • The convergence without regularity assumptions provides a constructive proof of existence.

Where Pith is reading between the lines

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

  • This method of proving existence via numerical convergence could extend to other coupled electromagnetic-thermal problems.
  • Bound-preserving techniques may improve approximations in related nonlinear PDE systems with low regularity.
  • Further work could explore time-dependent versions or three-dimensional extensions using similar justifications.

Load-bearing premise

A priori regularity results for the PDEs are available to justify the natural weak formulation of the problem.

What would settle it

A counterexample in a convex domain with a suitable mesh where the standard Galerkin finite element method fails to converge for this induction heating problem would disprove the convergence claim.

read the original abstract

In this paper, we analyse the existence of solutions and finite element approximation of a steady-state two-dimensional induction heating problem. One of the main difficulties of the problem is its right-hand side which, at a first sight, is only integrable. Using a priori regularity results for the PDEs involved it is shown that the natural weak formulation of the problem can be justified. Then, we study the finite element approximation and prove that the standard Galerkin FEM converges in convex domains and under suitable conditions on the mesh. We improve on this result by applying the recently-proposed bound-preserving method (BPM) to the heat equation, and show that this method converges to a solution of the problem under less stringent conditions on the domain and the mesh. As these analyses are carried out without any assumption on regularity of the solutions, then the convergence of the finite element method also proves existence of solutions. Several numerical experiments confirm the theoretical results, and showcase the improvement provided by the use of the bound-preserving method over the standard finite element method.

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 / 0 minor

Summary. The manuscript analyzes existence of solutions and finite element approximation for a steady-state two-dimensional induction heating problem whose right-hand side is only integrable. It first invokes a priori regularity results for the PDEs to justify that the natural weak formulation is well-defined, then proves convergence of the standard Galerkin FEM in convex domains under suitable mesh conditions and improved convergence of a bound-preserving method (BPM) under weaker domain and mesh assumptions. The authors conclude that, because the analyses impose no regularity assumptions on the solutions, FEM convergence also establishes existence.

Significance. If the justification of the weak form is non-circular and the convergence statements hold, the work supplies a route to existence via approximation for this nonlinear problem class, which is of interest in numerical analysis of induction heating models. The explicit comparison of standard Galerkin and BPM, together with the numerical experiments, provides concrete evidence of the practical improvement offered by the bound-preserving approach.

major comments (1)
  1. [Abstract] Abstract: The claim that 'these analyses are carried out without any assumption on regularity of the solutions' and that 'the convergence of the finite element method also proves existence of solutions' sits in tension with the preceding sentence that invokes a priori regularity results to justify the natural weak formulation. Standard a priori regularity theorems for the elliptic/parabolic systems involved typically require existence of a solution in a stronger space before higher integrability or differentiability can be derived; if those results are used to set up the continuous problem to which the discrete solutions converge, the existence argument is no longer independent of the regularity step.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading and for identifying a potential source of ambiguity in the abstract. We respond to the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that 'these analyses are carried out without any assumption on regularity of the solutions' and that 'the convergence of the finite element method also proves existence of solutions' sits in tension with the preceding sentence that invokes a priori regularity results to justify the natural weak formulation. Standard a priori regularity theorems for the elliptic/parabolic systems involved typically require existence of a solution in a stronger space before higher integrability or differentiability can be derived; if those results are used to set up the continuous problem to which the discrete solutions converge, the existence argument is no longer independent of the regularity step.

    Authors: We appreciate the referee's observation. The a priori regularity results cited are standard estimates for the linear elliptic equation (magnetic vector potential) and the linear parabolic heat equation, each with an L^1 right-hand side; these results establish higher integrability of the solutions from the data alone and do not rely on the existence of a solution to the coupled nonlinear system. Their sole purpose is to confirm that every term in the natural weak formulation is well-defined. The convergence proofs for both the standard Galerkin method and the bound-preserving method are then carried out directly on this weak form, without any further regularity hypotheses on the nonlinear solution. The limit of the discrete solutions therefore yields existence independently of the regularity step used only for justification of the formulation. We agree that the abstract wording could be tightened to make this separation explicit and will revise the abstract in the resubmission. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external results

full rationale

The paper justifies the weak formulation via external a priori regularity results for the involved PDEs and applies standard Galerkin theory to prove FEM convergence in convex domains. It explicitly states that analyses are performed without regularity assumptions on solutions, allowing convergence to also establish existence. No quoted step reduces a claimed prediction or existence result to a fitted input, self-definition, or load-bearing self-citation chain by construction. The argument is self-contained against the cited external regularity theorems and Galerkin framework.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on external a priori regularity results for the coupled PDEs; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption a priori regularity results for the PDEs involved
    Invoked to justify the natural weak formulation despite the RHS being only integrable.

pith-pipeline@v0.9.1-grok · 5711 in / 1158 out tokens · 24950 ms · 2026-06-25T23:14:58.037334+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

36 extracted references

  1. [1]

    I. A. Baratta, J. P. Dean, J. S. Dokken, M. Habera, J. S. Hale, C. N. Richardson, M. E. Rognes, M. W. Scroggs, N. N. Sime, and G.N. Wells. Dolfinx: The next generation fenics problem solving environ- ment,.Preprint, 2023

    2023

  2. [2]

    G. R. Barrenechea, E. H. Georgoulis, T. Pryer, and A. Veeser. A nodally bound-preserving finite element method.IMA Journal of Numerical Analysis, 44(4):2198–2219, 2024

    2024

  3. [3]

    G. R. Barrenechea, V. John, and P. Knobloch.Monotone Discretizations for Elliptic Second Order PartialDifferentialEquations,volume61ofSpringerSeriesonComputationalMathematics. Springer Cham, 2025

    2025

  4. [4]

    F. Bay, V. Labbe, Y. Favennec, and J. L. Chenot. A numerical model for induction heating processes coupling electromagnetism and thermomechanics.International Journal for Numerical Methods in Engineering, 58(6):839–867, 2003

    2003

  5. [5]

    North-HollandPublishingCo.,Amsterdam-NewYork,

    A.BensoussanandJ.-L.Lions.Applicationsofvariationalinequalitiesinstochasticcontrol,volume12 ofStudiesinMathematicsanditsApplications. North-HollandPublishingCo.,Amsterdam-NewYork,

  6. [6]

    Translated from the French

  7. [7]

    S. C. Brenner and L. R. Scott.The mathematical theory of finite element methods, volume 15 ofTexts in Applied Mathematics. Springer, 3rd edition, 2008. 3rd ed

    2008

  8. [8]

    Brezis.Functional analysis, Sobolev spaces and partial differential equations

    H. Brezis.Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011

    2011

  9. [9]

    Brezzi, W.W

    F. Brezzi, W.W. Hager, and P.-A. Raviart. Error estimates for the finite element solution of variational inequalities.Numer. Math., 28(4):431–443, 1977

    1977

  10. [10]

    Chaboudez, S

    C. Chaboudez, S. Clain, R. Glardon, J. Rappaz, M. Swierkosz, and R. Touzani. Numerical modelling of induction heating of long workpieces.IEEE Transactions on Magnetics, 30(6):5028–5037, 1994

    1994

  11. [11]

    Clain and R

    S. Clain and R. Touzani. Solution of a two-dimensional stationary induction heating problem without boundedness of the coefficients.Mathematical Modelling and Numerical Analysis, 31(7):845–870, 1997

    1997

  12. [12]

    Clain and R

    S. Clain and R. Touzani. A two-dimensional stationary induction heating problem.Mathematical Methods in the Applied Sciences, 20(9):759–766, 1997

    1997

  13. [13]

    Bestapproximationpropertyinthe𝑊 1 ∞ norm for finite element methods on graded meshes.Math

    A.Demlow,D.Leykekhman,A.H.Schatz,andL.B.Wahlbin. Bestapproximationpropertyinthe𝑊 1 ∞ norm for finite element methods on graded meshes.Math. Comp., 81(278):743–764, 2012. INDUCTION HEATING PROBLEM 29

    2012

  14. [14]

    Diening, J

    L. Diening, J. Rolfes, and A.J. Salgado. Pointwise gradient estimate of the Ritz projection.SIAM J. Numer. Anal., 62(3):1212–1225, 2024

    2024

  15. [15]

    Ern and J

    A. Ern and J. L. Guermond.Finite elements I - Approximation and interpolation, volume 72 ofTexts in Applied Mathematics. Springer, 2021

    2021

  16. [16]

    Ern and J

    A. Ern and J. L. Guermond.Finite Elements II - Galerkin Approximation, Elliptic and Mixed PDEs, volume 73 ofTexts in Applied Mathematics. Springer, 2021

    2021

  17. [17]

    R.S. Falk. Error estimates for the approximation of a class of variational inequalities.Math. Comput., 28:963–971, 1974

    1974

  18. [18]

    Friedman.Variational principles and free-boundary problems

    A. Friedman.Variational principles and free-boundary problems. Robert E. Krieger Publishing Co., Inc., Malabar, FL, second edition, 1988

    1988

  19. [19]

    Gmsh: A3-dfiniteelementmeshgeneratorwithbuilt-inpre-andpost- processingfacilities.InternationalJournalforNumericalMethodsinEngineering,79(11):1309–1331, 2009

    C.GeuzaineandJ.F.Remacle. Gmsh: A3-dfiniteelementmeshgeneratorwithbuilt-inpre-andpost- processingfacilities.InternationalJournalforNumericalMethodsinEngineering,79(11):1309–1331, 2009

    2009

  20. [20]

    Holst, M.G

    M.J. Holst, M.G. Larson, A. Målqvist, and R. Söderlund. Convergence analysis of finite element ap- proximations of the joule heating problem in three spatial dimensions.BIT Numerical Mathematics, 50(4):781–795, Dec 2010

    2010

  21. [21]

    Jensen and A

    M. Jensen and A. Målqvist. Finite element convergence for the joule heating problem with mixed boundary conditions.Bit Numerical Mathematics, 53(2):475–496, 2012

    2012

  22. [22]

    Jensen, A

    M. Jensen, A. Målqvist, and A. Persson. Finite element convergence for the time-dependent joule heating problem with mixed boundary conditions.IMA Journal of Numerical Analysis, 42(1):199, 2022

    2022

  23. [23]

    Jerison and C

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

    1995

  24. [24]

    DavidKinderlehrerandGuidoStampacchia.Anintroductiontovariationalinequalitiesandtheirappli- cations,volume31ofClassicsinappliedmathematics.SocietyforIndustrialandAppliedMathematics, 2000

    2000

  25. [25]

    A. F. D. Loula and J. Zhu. Mixed finite element analysis of a thermally nonlinear coupled problem. NumericalMethodsforPartialDifferentialEquations.AnInternationalJournal,22(1):180–196,2006

    2006

  26. [26]

    MacKenzie

    K. MacKenzie. Induction heating.https://github.com/Katie795/induction_heating, 2025

    2025

  27. [27]

    Afiniteelementpredictioncorrectionschemeformagneto-thermal coupled problem during curie transition.IEEE Transactions on Magnetics, 21(5):1871–1873, 1985

    P.Massé,B.Morel,andT.Breville. Afiniteelementpredictioncorrectionschemeformagneto-thermal coupled problem during curie transition.IEEE Transactions on Magnetics, 21(5):1871–1873, 1985

    1985

  28. [28]

    Nochetto, E

    R.H. Nochetto, E. Otárola, and A.J. Salgado. Convergence rates for the classical, thin and fractional elliptic obstacle problems.Philos. Trans. Roy. Soc. A, 373(2050):20140449, 14, 2015

    2050

  29. [29]

    Aquasi-statictwo-dimensionalinductionheatingproblem.PartI:modelling and analysis.Mathematical Models and Methods in Applied Sciences, 8(6):1003–1021, 1998

    C.PariettiandJ.Rappaz. Aquasi-statictwo-dimensionalinductionheatingproblem.PartI:modelling and analysis.Mathematical Models and Methods in Applied Sciences, 8(6):1003–1021, 1998

    1998

  30. [30]

    Aquasi-statictwo-dimensionalinductionheatingproblem.PartII:numerical analysis.Math Models and Methods in Applied Sciences, 9(9 ):1333–1350, 1999

    C.PariettiandJ.Rappaz. Aquasi-statictwo-dimensionalinductionheatingproblem.PartII:numerical analysis.Math Models and Methods in Applied Sciences, 9(9 ):1333–1350, 1999

    1999

  31. [31]

    Rannacher and R

    R. Rannacher and R. Scott. Some optimal error-estimates for piecewise linear finite-element approxi- mations.Mathematics of Computation, 38(158):437–445, 1982

    1982

  32. [32]

    P. A. Raviart and J. M. Thomas.Introduction à l’analyse numérique des équations aux dérivées par- tielles. Collection mathématiques appliquées pur la maîtrise sous la direction de P. G. Ciarlet et J. L. Lions. Masson, 1992

    1992

  33. [33]

    North-HollandPublishingCo., Amsterdam, 1987

    J.-F.Rodrigues.Obstacleproblemsinmathematicalphysics,volume134ofNorth-HollandMathemat- icsStudies. North-HollandPublishingCo., Amsterdam, 1987. NotasdeMatemática, 114.[Mathemat- ical Notes]

    1987

  34. [34]

    Roubí̆cek.Nonlinear partial differential equations with applications, volume 153 ofInternational Series of Numerical Mathematics

    T. Roubí̆cek.Nonlinear partial differential equations with applications, volume 153 ofInternational Series of Numerical Mathematics. Birkhäuser, 2nd. edition, 2013

    2013

  35. [35]

    Scientific Computation

    R.TouzaniandJ.Rappaz.MathematicalModelsforEddyCurrentsandMagnetostatics: WithSelected Applications. Scientific Computation. Springer, 2014. 30 G.R. BARRENECHEA, K. MACKENZIE, AND A.J. SALGADO

    2014

  36. [36]

    MixeddiscontinuousGalerkinanalysisofthermallynonlinearcoupled problem.Computer Methods in Applied Mechanics and Engineering, 200(13):1479–1489, 2011

    J.Zhu,X.Yu,andA.F.D.Loula. MixeddiscontinuousGalerkinanalysisofthermallynonlinearcoupled problem.Computer Methods in Applied Mechanics and Engineering, 200(13):1479–1489, 2011. (G.R. Barrenechea) DEPARTMENT OFMATHEMATICS ANDSTATISTICS, UNIVERSITY OFSTRATHCLYDE, 26 RICHMOND STREET, GLASGOW, G1 1XH UNITEDKINGDOM Email address:gabriel.barrenechea@strath.ac.u...

    2011