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arxiv: 2604.14539 · v1 · submitted 2026-04-16 · 🧮 math.NA · cs.NA

Spurious-mode-free finite element method for scattering resonances in transmission problems

Bo Gong , Jiguang Sun This is my paper

Pith reviewed 2026-05-10 11:05 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords scattering resonancestransmission problemsfinite element methodDirichlet-to-Neumann mapholomorphic Fredholm operatorspurious modesnonlinear eigenvalue problemsspectrum indicator method
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The pith

Finite element method with Dirichlet-to-Neumann truncation computes scattering resonances without spurious modes and with optimal convergence.

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

The paper develops a numerical method to find scattering resonances in transmission problems that avoids the spurious modes plaguing other approaches. It truncates the infinite domain using an exact Dirichlet-to-Neumann map and recasts the resonances as eigenvalues of a holomorphic Fredholm operator. This operator is then discretized using standard finite elements, turning the problem into a nonlinear matrix eigenvalue problem solved via the spectrum indicator method. The authors prove that this discretization converges at the optimal rate. A sympathetic reader would care because reliable resonance computations matter for understanding wave behavior in applications like optics, acoustics, and quantum mechanics, where fake modes can lead to incorrect predictions.

Core claim

We propose a spurious-mode-free method for computing scattering resonances in transmission problems. The unbounded domain is truncated using a Dirichlet-to-Neumann map. The resonances are formulated as eigenvalues of a holomorphic Fredholm operator function, which is discretized by the finite element method. The spectrum indicator method is then used to compute the eigenvalues of the nonlinear matrix eigenvalue problems. We establish optimal order convergence and present extensive examples that demonstrate the effectiveness of the proposed method.

What carries the argument

Holomorphic Fredholm operator function obtained from Dirichlet-to-Neumann truncation of the transmission problem, discretized by finite elements and solved via the spectrum indicator method for the resulting nonlinear eigenvalue problem.

Load-bearing premise

The combination of the Dirichlet-to-Neumann map, finite element discretization, and spectrum indicator method eliminates all spurious modes and achieves optimal order convergence without additional assumptions on geometry or material properties.

What would settle it

Applying the method to a transmission problem with known exact resonance frequencies and checking whether spurious modes appear near the true values or whether the computed error decays at a rate below the predicted optimal order would settle the claim.

read the original abstract

Scattering resonances arise in wave phenomena and play an important role in many applications. While extensive theoretical studies have been conducted, effective numerical computation remains limited, and most existing methods suffer from spurious modes. In this paper, we propose a spurious-mode-free method for computing scattering resonances in transmission problems. The unbounded domain is truncated using a Dirichlet-to-Neumann (DtN) map. The resonances are formulated as eigenvalues of a holomorphic Fredholm operator function, which is discretized by the finite element method. The spectrum indicator method is then used to compute the eigenvalues of the nonlinear matrix eigenvalue problems. We establish optimal order convergence and present extensive examples that demonstrate the effectiveness of the proposed method. The results are consistent with existing theoretical findings in the literature and offer new insights that may inform further theoretical developments.

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 proposes a spurious-mode-free finite element method for computing scattering resonances in transmission problems. The unbounded domain is truncated via a Dirichlet-to-Neumann map, resonances are cast as eigenvalues of a holomorphic Fredholm operator function, the operator is discretized by FEM, and the spectrum indicator method is applied to the resulting nonlinear matrix eigenvalue problem. The authors claim optimal-order convergence of the method and demonstrate its effectiveness through numerical examples that align with existing theory.

Significance. If the central claims hold, the work would provide a reliable, spurious-mode-free numerical tool for an important class of resonance problems in wave scattering, where existing methods often suffer from artifacts. The formulation via holomorphic Fredholm operators combined with the spectrum indicator method is a strength, as is the explicit truncation strategy. However, the significance hinges on whether the convergence analysis truly achieves optimal rates without hidden regularity assumptions on the interfaces or coefficients, which is a common limitation in transmission problems.

major comments (2)
  1. [Abstract and convergence analysis] Abstract and convergence analysis section: the claim of optimal-order convergence is asserted without extra assumptions on geometry or material properties, yet transmission problems with discontinuous coefficients or non-smooth interfaces typically yield only reduced FEM rates (e.g., O(h^{1/2}) in H^1). The proof must explicitly show how the DtN truncation, discrete Fredholm property, and spectrum indicator together preserve optimal rates and eliminate spurious modes independently of such regularity; this is load-bearing for the central claim.
  2. [Discrete operator and spectrum indicator] Section on the discrete operator and spectrum indicator: it is not clear from the formulation whether the artificial boundary truncation and FEM discretization inherit spectral isolation without introducing pollution near the continuous spectrum. A concrete error estimate or numerical test isolating the effect of the DtN approximation on eigenvalue accuracy is needed to support the spurious-mode-free assertion.
minor comments (2)
  1. [Introduction] Notation for the holomorphic operator function and its discretization should be introduced with explicit reference to the underlying transmission problem to improve readability.
  2. [Numerical examples] The numerical examples section would benefit from a table comparing computed resonances against known analytical or reference values, including error norms and observed convergence rates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the presentation of our results. We address each major comment below and will incorporate revisions to strengthen the paper.

read point-by-point responses

  1. Referee: [Abstract and convergence analysis] Abstract and convergence analysis section: the claim of optimal-order convergence is asserted without extra assumptions on geometry or material properties, yet transmission problems with discontinuous coefficients or non-smooth interfaces typically yield only reduced FEM rates (e.g., O(h^{1/2}) in H^1). The proof must explicitly show how the DtN truncation, discrete Fredholm property, and spectrum indicator together preserve optimal rates and eliminate spurious modes independently of such regularity; this is load-bearing for the central claim.

    Authors: We appreciate the referee pointing out the need for explicit clarification on regularity. Our convergence analysis (Section 4) establishes optimal-order convergence in the sense of the best rate permitted by the Sobolev regularity of the eigenfunctions, which depends on the smoothness of the interfaces and coefficients; this is the standard notion of optimality for transmission problems. The proof shows that the DtN truncation introduces an error that is exponentially small in the truncation radius and does not degrade the FEM rate, while the discrete operator remains a holomorphic Fredholm operator of index zero. The spectrum indicator then isolates the discrete eigenvalues corresponding to physical resonances. We will revise the abstract and add a remark in the convergence section to state the regularity assumptions explicitly and detail how the combination of DtN, FEM, and spectrum indicator preserves the rate without hidden assumptions beyond those standard for the problem class. revision: yes

  2. Referee: [Discrete operator and spectrum indicator] Section on the discrete operator and spectrum indicator: it is not clear from the formulation whether the artificial boundary truncation and FEM discretization inherit spectral isolation without introducing pollution near the continuous spectrum. A concrete error estimate or numerical test isolating the effect of the DtN approximation on eigenvalue accuracy is needed to support the spurious-mode-free assertion.

    Authors: We agree that an explicit demonstration of the DtN effect would strengthen the spurious-mode-free claim. The continuous operator is holomorphic and Fredholm away from the continuous spectrum; the DtN truncation preserves holomorphy in a neighborhood of the resonances of interest, and the conforming FEM discretization inherits the Fredholm property. To provide concrete support, we will add a new numerical test in the revised manuscript that isolates the truncation error by computing a fixed resonance for increasing artificial boundary radii and reporting the eigenvalue error decay, together with a brief perturbation estimate showing the DtN contribution is negligible relative to the FEM discretization error. revision: yes

Circularity Check

0 steps flagged

No circularity; standard operator-theoretic formulation and FEM analysis are self-contained.

full rationale

The derivation begins with the standard truncation of the exterior domain via the Dirichlet-to-Neumann map, formulates resonances as eigenvalues of a holomorphic Fredholm operator-valued function, applies conforming finite-element discretization, and invokes the spectrum indicator method for nonlinear eigenproblems. Optimal-order convergence is asserted via classical a priori estimates for the discrete operator and the isolation properties of the spectrum indicator; none of these steps are shown to reduce by definition or by self-citation to the target quantities themselves. The paper cites external literature for the underlying Fredholm theory and does not rely on fitted parameters renamed as predictions or on uniqueness theorems imported solely from the authors' prior work. The numerical examples serve only as verification, not as the source of the claimed rates.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of holomorphic Fredholm operators and FEM convergence theory for eigenvalue problems, which are assumed from prior literature without new free parameters or invented entities detailed in the abstract.

axioms (2)
  • standard math Holomorphic Fredholm operator functions can be used to formulate scattering resonances as eigenvalues
    Invoked when reformulating the resonance problem.
  • domain assumption Finite element discretization of the truncated problem achieves optimal convergence
    Stated as established but not proven in the provided abstract.

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Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [1]

    Braess, Finite Elements

    D. Braess, Finite Elements. Theory, Fast Solvers, and Applications in Elasticity Theory. 3rd ed., Cambridge University Press, Cambridge, 2007

    work page 2007

  2. [2]

    S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods. 3rd ed., Texts in Applied Mathematics, 15. Springer, New York, 2008

    work page 2008

  3. [3]

    P. G. Ciarlet and P.-A. Raviart,The combined effect of curved boundaries and numerical in- tegration in isoparametric finite element methods.in The mathematical foundations of the finite element method with applications to partial differential equations, 409-474, Academic Press, New York-London, 1972

    work page 1972

  4. [4]

    Dyatlov and M

    S. Dyatlov and M. Zworski, Mathematical Theory of Scattering Resonances. American Mathe- matical Society, Providence, RI, 2019

    work page 2019

  5. [5]

    Dubios,Discrete vector potential representation of a divergence-free vector field in three- dimensional domains: numerical analysis of a model problem.SIAM J

    F. Dubios,Discrete vector potential representation of a divergence-free vector field in three- dimensional domains: numerical analysis of a model problem.SIAM J. Numer. Anal. 27 (1990), no. 5, 1103-1141

    work page 1990

  6. [6]

    O. G. Ernst,A finite-element capacitance matrix method for exterior Helmholtz problems.Numer. Math. 75 (1996), no. 2, 175-204

    work page 1996

  7. [7]

    Galkowski,The quantum Sabine law for resonances in transmission problems.Pure Appl

    J. Galkowski,The quantum Sabine law for resonances in transmission problems.Pure Appl. Anal. 1 (2019), no. 1, 27-100

    work page 2019

  8. [8]

    B. Gong, T. Sato, J. Sun, and X. Wu,FEM-DtN-SIM method for computing resonances of Schr¨ odinger operators.Appl. Math. Lett. 173 (2026), 109803

    work page 2026

  9. [9]

    Grubiˇ si´ c, R

    L. Grubiˇ si´ c, R. Hiptmair, and D. Renner,Detecting near resonances in acoustic scattering.J. Sci. Comput. 96 (2023), no. 3, Paper No. 81

    work page 2023

  10. [10]

    G. J. Fix,Effects of quadrature errors in finite element approximation of steady state, eigenvalue and parabolic problems.in The mathematical foundations of the finite element method with applications to partial differential equations, 525-556, Academic Press, New York-London, 1972

    work page 1972

  11. [11]

    Halla,Radial complex scaling for anisotropic scalar resonance problems.SIAM J

    M. Halla,Radial complex scaling for anisotropic scalar resonance problems.SIAM J. Numer. Anal. 60 (2022), no. 5, 2713-2730

    work page 2022

  12. [12]

    Hiptmair, A

    R. Hiptmair, A. Moiola, and E. A. Spence,Spurious quasi-resonances in boundary integral equations for the Helmholtz transmission problem.SIAM J. Appl. Math. 82(2022), no. 4, 1446–1469

    work page 2022

  13. [13]

    Hohage and L

    T. Hohage and L. Nannen,Hardy space infinite elements for scattering and resonance problems. SIAM J. Numer. Anal. 47 (2009), no. 2, 972-996

    work page 2009

  14. [14]

    G. C. Hsiao, N. Nigam, J. E. Pasciak, and L. Xu,Error analysis of the DtN-FEM for the scattering problem in acoustics via Fourier analysis.J. Comput. Appl. Math. 235 (2011), no. 17, 4949-4965

    work page 2011

  15. [15]

    Karma,Approximation in eigenvalue problems for holomorphic Fredholm operator functions I.Numer

    O. Karma,Approximation in eigenvalue problems for holomorphic Fredholm operator functions I.Numer. Funct. Anal. Optim. 17 (1996), no. 3-4, 365-387

    work page 1996

  16. [16]

    Karma,Approximation in eigenvalue problems for holomorphic Fredholm operator functions II (convergence rate).Numer

    O. Karma,Approximation in eigenvalue problems for holomorphic Fredholm operator functions II (convergence rate).Numer. Funct. Anal. Optim. 17 (1996), no. 3-4, 389-408. 18

    work page 1996

  17. [17]

    Kim and J

    S. Kim and J. Pasciak,The computation of resonances in open systems using a perfectly matched layer.Math. Comp. 78 (2009), no. 267, 1375-1398

    work page 2009

  18. [18]

    P. D. Lax and R. S. Phillips, Scattering Theory. Pure Appl. Math. 26. Academic Press, Boston, 1989

    work page 1989

  19. [19]

    Lenoir, M

    M. Lenoir, M. Vullierme-Ledard and C. Hazard.Variational formulations for the determination of resonant states in scattering problems.SIAM J. Math. Anal. 23 (1992), no. 3, 579-608

    work page 1992

  20. [20]

    X. Liu, J. Sun and L. Zhang,Accurate computation of scattering poles of acoustic obstacles with impedance boundary conditions.Wave Motion 132 (2025), 103425

    work page 2025

  21. [21]

    Ma and J

    Y. Ma and J. Sun,Computation of scattering poles using boundary integrals.Appl. Math. Lett. 146 (2023), 108792

    work page 2023

  22. [22]

    Matsushima and T

    K. Matsushima and T. Yamada,Exceptional points and defective resonances in an acoustic scattering system with sound-hard obstacles.Proc. A 481 (2025), no. 2320, 20250230

    work page 2025

  23. [23]

    Misawa, K

    R. Misawa, K. Niino, and N. Nishimura,Boundary integral equations for calculating complex eigenvalues of transmission problems.SIAM J. Appl. Math. 77 (2017), 770-788

    work page 2017

  24. [24]

    Moiola and E

    A. Moiola and E. A. Spence,Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions.Math. Models Methods Appl. Sci. 29 (2019), no. 2, 317-354

    work page 2019

  25. [25]

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

    work page 2003

  26. [26]

    Nannen and M

    L. Nannen and M. Wess,Computing scattering resonances using perfectly matched layers with frequency dependent scaling functions.BIT 58 (2018), no. 2, 373-395

    work page 2018

  27. [27]

    Popov and Vodev,Resonances near the real axis for transparent obstacles.Comm

    G. Popov and Vodev,Resonances near the real axis for transparent obstacles.Comm. Math. Phys. 207 (1999), no. 2, 411-438

    work page 1999

  28. [28]

    Steinbach and G

    O. Steinbach and G. Unger,Convergence analysis of a Galerkin boundary element method for the Dirichlet Laplacian eigenvalue problem.SIAM J. Numer. Anal. 50 (2012), no. 2, 710-728

    work page 2012

  29. [29]

    Steinbach and G

    O. Steinbach and G. Unger,Combined boundary integral equations for acoustic scattering- resonance problems.Math. Methods Appl. Sci. 40 (2017), no. 5, 1516-1530

    work page 2017

  30. [30]

    Sun and A

    J. Sun and A. Zhou, Finite Element Methods for Eigenvalue Problems, Chapman and Hall/CRC, Boca Raton, FL, 2016

    work page 2016

  31. [31]

    M. E. Taylor, Partial differential equations. II. Qualitative studies of linear equations. Applied Mathematical Sciences, 116. Springer-Verlag, New York, 1996

    work page 1996

  32. [32]

    ¨Uberall, et al.,Electromagnetic and acoustic resonance scattering theory.Wave Motion 5 (1983), no

    H. ¨Uberall, et al.,Electromagnetic and acoustic resonance scattering theory.Wave Motion 5 (1983), no. 4, 307-329

    work page 1983

  33. [33]

    Vainikko

    G. Vainikko. Funktionalanalysis der diskretisierungsmethoden. Mit Englischen und Russischen Zusammenfassungen. Teubner-Texte zur Mathematik. B. G. Teubner Verlag, Leipzig, 1976

    work page 1976

  34. [34]

    Y. Xi, B. Gong, and J. Sun,Analysis of a finite element DtN method for scattering resonances of sound hard obstacles.arXiv:2404.09300, 2024

    work page arXiv 2024

  35. [35]

    Y. Xi, J. Lin, and J. Sun,A finite element contour integral method for computing the resonances of metallic grating structures with subwavelength holes.Comput. Math. Appl. 170 (2024), 161-171

    work page 2024

  36. [36]

    Xi and X

    Y. Xi and X. Ji,A finite element contour integral method for computing the scattering reso- nances of fluid-solid interaction problem.J. Comput. Phys. 521 (2025), part 1, Paper No. 113539, 15 pp

    work page 2025

  37. [37]

    Xi and J

    Y. Xi and J. Sun,Parallel multi-step contour integral methods for nonlinear eigenvalue prob- lems.arXiv:2312.13117, 2023

    work page arXiv 2023

  38. [38]

    Zl´ amal,Curved elements in the finite element method

    M. Zl´ amal,Curved elements in the finite element method. I.SIAM J. Numer. Anal. 10 (1973), no. 1, 229-240

    work page 1973

  39. [39]

    Zworski,Resonances in physics and geometry.Notices Amer

    M. Zworski,Resonances in physics and geometry.Notices Amer. Math. Soc. 46 (1999), no. 3, 319-328. 19

    work page 1999