pith. sign in

arxiv: 2511.20103 · v2 · pith:FI2PILEInew · submitted 2025-11-25 · 🧮 math.NA · cs.NA

Multiscale Methods for wave propagation in materials with sign-changing coefficients

Eric T. Chung , Patrick Ciarlet Jr. , Xingguang Jin , Changqing Ye This is my paper

Pith reviewed 2026-05-22 12:33 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords CEM-GMsFEMsign-changing coefficientsT-coercivityinf-sup stabilitymultiscale finite elementselectromagnetic wavesmetamaterialsa priori estimates
0
0 comments X

The pith

Multiscale finite element method achieves stability for waves in sign-changing media

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

This paper modifies the Constraint Energy Minimizing Generalized Multiscale Finite Element Method to solve time-harmonic electromagnetic wave equations when the material coefficients can change sign. The modification involves tailoring auxiliary spaces and applying T-coercivity theory with resolution conditions to prove inf-sup stability and derive error estimates. Readers interested in metamaterial modeling would care because negative coefficients violate the usual coercivity assumptions and can make standard numerical methods unreliable. The presented numerical results indicate that the adapted method handles such cases effectively.

Core claim

The adapted CEM-GMsFEM, with auxiliary spaces constructed to fit the sign-changing setting, satisfies the inf-sup condition under appropriate resolution conditions according to T-coercivity theory, and this yields an a priori error estimate for the approximation of the wave solution.

What carries the argument

Tailored auxiliary spaces in the CEM-GMsFEM that restore inf-sup stability for non-coercive operators arising from sign-changing coefficients.

If this is right

  • The discrete problem remains well-posed when resolution conditions hold.
  • An error bound controls the difference between the discrete and continuous solutions.
  • The method applies to electromagnetic problems featuring negative dielectric permittivity or magnetic permeability.
  • Robust performance is observed in numerical tests with complex sign-changing coefficient distributions.

Where Pith is reading between the lines

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

  • Similar adaptations might apply to other wave problems involving non-coercive operators, such as certain acoustic models.
  • Checking the resolution conditions during computation could serve as a practical reliability indicator.
  • Extensions to time-dependent or nonlinear problems would require additional analysis of the same stability framework.

Load-bearing premise

The existence of auxiliary spaces that meet the resolution conditions necessary to maintain inf-sup stability despite sign changes in the coefficients.

What would settle it

Run the method on a problem where the resolution condition is not satisfied and observe whether the linear system loses uniqueness or the error exceeds the predicted bound.

Figures

Figures reproduced from arXiv: 2511.20103 by Changqing Ye, Eric T. Chung, Patrick Ciarlet Jr., Xingguang Jin.

Figure 1
Figure 1. Figure 1: An illustration of the two-scale mesh, a fine element [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The relative errors of the proposed method with different numbers of over [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) The solution approximated by the CEM-GMsFEM when we choose H = 1/40 and m = 3. (d) The solution approximated by the Q1 FEM when we choose H = 1/40. (d) The solution approximated by the Q1 FEM when we choose H = 1/40. (b)/(e) The exact solution. (c) The difference of the solution approximated by CEM￾GMsFEM with the exact solution. (f) The difference of solution approximated by Q1-FEM with exact solution… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Coefficients with (σ + ∗ , σ− ∗ ) = (1, 103 ); (b) Source term based on Gaussian functions. and Ω− are demonstrated in [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The relative errors of the proposed method with different numbers of over [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) The solution approximated by the CEM-GMsFEM when we choose H = 1/40 and m = 3. (b) The solution approximated by the Q1 FEM when we choose H = 1/40 [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The relative errors of the proposed method with different numbers of basis [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) Gaussian Beam Source centered at (0, 0.5). (b) The Negative index Materials [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The relative errors of the proposed method with different numbers of over [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a) The solution approximated by the CEM-GMsFEM when we choose H = 1/40 and m = 3. (b) The solution approximated by the Q1 FEM when we choose H = 1/40 [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: (a) The solution approximated by the CEM-GMsFEM when we choose H = 1/40 and m = 3. (b) The solution approximated by the Q1 FEM when we choose H = 1/40. 6. Conclusion. We have proposed a multiscale computational method for solv￾ing conventional electromagnetic wave problems, utilizing the framework of CEM￾GMsFEMs in the construction of multiscale basis functions. Due to a non-positive definite right-hand b… view at source ↗
read the original abstract

From a mathematical perspective, the extraordinary properties of metamaterials are often reflected in the coefficients of the governing partial differential equations (PDEs). These coefficients may fall outside the assumptions of classical theory, particularly when the effective dielectric permittivity and/or magnetic permeability are negative. This situation can transform a coercive operator into a non-coercive one, potentially leading to ill-posedness. In this paper, we utilize the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM), specifically designed for time-harmonic electromagnetic wave problems, where the construction of auxiliary spaces in the original CEM-GMsFEM is tailored to accommodate the sign-changing setting. Based on the framework of \texttt{T}-coercivity theory and resolution conditions, we establish the inf-sup stability and provide an a priori error estimate for the proposed method. The numerical results demonstrate the effectiveness and robustness of our approach in handling such sophisticated coefficient profiles.

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 manuscript extends the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) to time-harmonic electromagnetic wave problems with sign-changing coefficients. It tailors the construction of auxiliary spaces to this setting and invokes T-coercivity theory together with resolution conditions to establish inf-sup stability and derive an a priori error estimate; numerical experiments are presented to illustrate robustness for selected coefficient profiles.

Significance. If the inf-sup constant can be shown to remain bounded independently of the negative contrast, the result would supply a practical multiscale tool for non-coercive wave problems arising in metamaterials, extending an existing CEM-GMsFEM framework with explicit stability and error control.

major comments (2)
  1. [§3.2, Theorem 3.1] §3.2, Theorem 3.1 (inf-sup stability): the argument combines T-coercivity with resolution conditions on the auxiliary spaces, yet the proof does not explicitly track the dependence of the inf-sup constant on the magnitude of the negative coefficient; if this constant deteriorates with increasing contrast, both the stability claim and the subsequent a priori error bound lose uniformity.
  2. [§4.1, Eq. (4.3)] §4.1, Eq. (4.3) (resolution condition): the condition is stated in terms of the mesh size and the support of the negative part, but no quantitative estimate is given showing that the constant in the T-coercivity inequality remains independent of the contrast ratio; this dependence is load-bearing for the claimed robustness.
minor comments (2)
  1. [Numerical experiments] The numerical section would benefit from an additional table or plot that varies the negative contrast over several orders of magnitude while reporting the observed inf-sup constant or error behavior.
  2. [§2.3] Notation for the auxiliary space construction in §2.3 could be aligned more closely with the original CEM-GMsFEM references to ease comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, with a focus on the dependence of stability constants on the contrast ratio.

read point-by-point responses

  1. Referee: [§3.2, Theorem 3.1] the argument combines T-coercivity with resolution conditions on the auxiliary spaces, yet the proof does not explicitly track the dependence of the inf-sup constant on the magnitude of the negative coefficient; if this constant deteriorates with increasing contrast, both the stability claim and the subsequent a priori error bound lose uniformity.

    Authors: We appreciate this observation. The proof of Theorem 3.1 applies T-coercivity at the continuous level and transfers stability to the discrete setting via the approximation properties of the auxiliary spaces, which are enforced by the resolution conditions. The inf-sup constant therefore inherits dependence on the contrast from the T-coercivity constant of the continuous problem. The manuscript does not derive an explicit uniform bound independent of contrast. Numerical experiments nevertheless indicate practical robustness. In the revised manuscript we will add a remark following Theorem 3.1 that explicitly notes this dependence and clarifies under which additional assumptions on the coefficient a uniform bound might hold. revision: partial

  2. Referee: [§4.1, Eq. (4.3)] the condition is stated in terms of the mesh size and the support of the negative part, but no quantitative estimate is given showing that the constant in the T-coercivity inequality remains independent of the contrast ratio; this dependence is load-bearing for the claimed robustness.

    Authors: We agree that an explicit quantitative link between the resolution condition (4.3) and contrast independence of the T-coercivity constant would strengthen the robustness statement. Equation (4.3) guarantees that the local auxiliary problems resolve the interfaces and oscillations associated with the negative part sufficiently well for the global error analysis to close. The T-coercivity constant itself is taken from the continuous problem and its contrast dependence is not quantified in the present analysis. We will revise the discussion in §4.1 to include a paragraph explaining the interaction between the resolution parameter and the T-coercivity framework, together with a note on the contrast dependence. revision: partial

Circularity Check

0 steps flagged

No circularity: stability derived from external T-coercivity framework

full rationale

The paper's central claims of inf-sup stability and a priori error estimates rest on applying the established T-coercivity theory together with resolution conditions to a modified CEM-GMsFEM construction for sign-changing coefficients. The abstract explicitly positions this as building on an external mathematical framework rather than fitting parameters inside the paper or reducing the result to a self-definition. No load-bearing step is shown to collapse by construction to an input quantity, self-citation, or renamed ansatz; the derivation therefore remains independent and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions of multiscale finite-element theory plus the external T-coercivity framework; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption T-coercivity theory provides a suitable framework for non-coercive problems arising from sign-changing coefficients
    Invoked to establish inf-sup stability for the discrete problem.
  • domain assumption Resolution conditions hold for the chosen auxiliary spaces in the sign-changing setting
    Required for the stability and error analysis to go through.

pith-pipeline@v0.9.0 · 5696 in / 1354 out tokens · 51897 ms · 2026-05-22T12:33:29.275153+00:00 · methodology

discussion (0)

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

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Numerical homogenization for indefinite time-harmonic Maxwell equations

    math.NA 2026-04 unverdicted novelty 6.0

    A new edge-multiscale homogenization method for high-wavenumber Maxwell equations in heterogeneous media achieves near-linear mesh-size scaling with wavenumber via a nonstandard variational formulation.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [2]

    Barr´e and P

    M. Barr´e and P. Ciarlet Jr. , The T-coercivity approach for mixed problems . In press in Comptes Rendus Mathematique, 2023, https://hal.science/hal-03820910

    work page 2023

  2. [3]

    Bonnet-Ben Dhia, P

    A. Bonnet-Ben Dhia, P. Ciarlet, and C. Zw ¨olf, Time harmonic wave diffraction problems in materials with sign-shifting coefficients , Journal of Computational and Applied Mathe- matics, 234 (2010), pp. 1912–1919, https://doi.org/https://doi.org/10.1016/j.cam.2009.08. 041

    work page doi:10.1016/j.cam.2009.08 2010

  3. [4]

    Bonnet-BenDhia, L

    A.-S. Bonnet-BenDhia, L. Chesnel, and P. Ciarlet Jr. , T-coercivity for scalar interface problems between dielectrics and metamaterials, ESAIM. Mathematical Modelling and Nu- merical Analysis, 46 (2012), pp. 1363–1387, https://doi.org/10.1051/m2an/2012006

    work page doi:10.1051/m2an/2012006 2012

  4. [5]

    Bonnet-BenDhia, L

    A.-S. Bonnet-BenDhia, L. Chesnel, and P. Ciarlet Jr., T-coercivity for the Maxwell prob- lem with sign-changing coefficients , Communications in Partial Differential Equations, 39 (2014), pp. 1007–1031, https://doi.org/10.1080/03605302.2014.892128

    work page doi:10.1080/03605302.2014.892128 2014

  5. [6]

    Bonnet-BenDhia, L

    A.-S. Bonnet-BenDhia, L. Chesnel, and P. Ciarlet Jr. , Two-dimensional Maxwell’s equa- tions with sign-changing coefficients , Applied Numerical Mathematics, 79 (2014), pp. 29– 41, https://doi.org/10.1016/j.apnum.2013.04.006

    work page doi:10.1016/j.apnum.2013.04.006 2014

  6. [7]

    Carvalho, L

    C. Carvalho, L. Chesnel, and P. Ciarlet Jr. , Eigenvalue problems with sign-changing coefficients, Comptes Rendus Mathematique, 355 (2017), pp. 671–675, https://doi.org/10. 1016/j.crma.2017.05.002

    work page 2017

  7. [8]

    Chaumont-Frelet and B

    Chaumont-Frelet, Th ´eophile and Verf ¨urth, Barbara , A generalized finite element method for problems with sign-changing coefficients , ESAIM: M2AN, 55 (2021), pp. 939– 967, https://doi.org/10.1051/m2an/2021007

    work page doi:10.1051/m2an/2021007 2021

  8. [9]

    Chung and C

    E. Chung and C. S. Lee , A staggered discontinuous galerkin method for the convec- tion–diffusion equation , Journal of Numerical Mathematics, 20 (2012), pp. 1–32, https: //doi.org/doi:10.1515/jnum-2012-0001

    work page doi:10.1515/jnum-2012-0001 2012

  9. [10]

    E. T. Chung and P. Ciarlet, A staggered discontinuous galerkin method for wave propagation in media with dielectrics and meta-materials, Journal of Computational and Applied Math- ematics, 239 (2013), pp. 189–207, https://doi.org/https://doi.org/10.1016/j.cam.2012.09. 033

    work page doi:10.1016/j.cam.2012.09 2013

  10. [11]

    E. T. Chung, Y. Efendiev, and W. T. Leung, Constraint energy minimizing generalized mul- This manuscript is for review purposes only. CEM-GMSFEM FOR SIGN-CHANGING 25 tiscale finite element method , Computer Methods in Applied Mechanics and Engineering, 339 (2018), pp. 298–319, https://doi.org/https://doi.org/10.1016/j.cma.2018.04.010

    work page doi:10.1016/j.cma.2018.04.010 2018

  11. [12]

    E. T. Chung, Y. Efendiev, and G. Li , An adaptive gmsfem for high-contrast flow problems , Journal of Computational Physics, 273 (2014), pp. 54–76, https://doi.org/https://doi.org/ 10.1016/j.jcp.2014.05.007

    work page doi:10.1016/j.jcp.2014.05.007 2014

  12. [13]

    Ciarlet Jr

    P. Ciarlet Jr. , T-coercivity: Application to the discretization of Helmholtz-like problems , Computers & Mathematics with Applications, 64 (2012), pp. 22–34, https://doi.org/10. 1016/j.camwa.2012.02.034

    work page 2012

  13. [14]

    T. Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media , Journal of Computational Physics, 134 (1997), pp. 169–189, https://doi.org/https://doi.org/10.1006/jcph.1997.5682

    work page doi:10.1006/jcph.1997.5682 1997

  14. [15]

    H. H. Kim, E. T. Chung, and C. S. Lee , A staggered discontinuous galerkin method for the stokes system , SIAM Journal on Numerical Analysis, 51 (2013), pp. 3327–3350, https: //doi.org/10.1137/120896037

    work page doi:10.1137/120896037 2013

  15. [16]

    J. Li, Y. Chen, and V. Elander , Mathematical and numerical study of wave propagation in negative-index materials, Computer Methods in Applied Mechanics and Engineering, 197 (2008), pp. 3976–3987, https://doi.org/https://doi.org/10.1016/j.cma.2008.03.017

    work page doi:10.1016/j.cma.2008.03.017 2008

  16. [17]

    Z. Li, C. Ye, and E. T. Chung , An iterative constraint energy minimizing generalized mul- tiscale finite element method for contact problem , Journal of Computational and Applied Mathematics, 461 (2025), p. 116398, https://doi.org/https://doi.org/10.1016/j.cam.2024. 116398

    work page doi:10.1016/j.cam.2024 2025

  17. [18]

    M˚alqvist and D

    A. M˚alqvist and D. Peterseim , Localization of elliptic multiscale problems , Mathemat- ics of Computation, 83 (2014), pp. 2583–2603, https://doi.org/https://doi.org/10.1090/ S0025-5718-2014-02868-8

    work page 2014

  18. [19]

    Pendry, A

    J. Pendry, A. Holden, D. Robbins, and W. Stewart , Magnetism from conductors and enhanced nonlinear phenomena, IEEE Transactions on Microwave Theory and Techniques, 47 (1999), pp. 2075–2084, https://doi.org/10.1109/22.798002

    work page doi:10.1109/22.798002 1999

  19. [20]

    J. B. Pendry, Negative refraction makes a perfect lens , Physical review letters, 85 18 (2000), pp. 3966–9, https://doi.org/https://doi.org/10.1103/physrevlett.85.3966

    work page doi:10.1103/physrevlett.85.3966 2000

  20. [21]

    L. A. Poveda, J. Galvis, and E. T. Chung , A second-order exponential integration con- straint energy minimizing generalized multiscale method for parabolic problems, Journal of Computational Physics, 502 (2024), p. 112796, https://doi.org/10.1016/j.jcp.2024.112796

    work page doi:10.1016/j.jcp.2024.112796 2024

  21. [22]

    R. A. Shelby, D. R. Smith, and S. Schultz , Experimental verification of a negative index of refraction, Science, 292 (2001), pp. 77 – 79, https://doi.org/10.1126/science.1058847

    work page doi:10.1126/science.1058847 2001

  22. [23]

    V. G. Veselago , The Electrodynamics of Substances with Simultaneously Negative Values of ε and µ, Soviet Physics Uspekhi, 10 (1968), pp. 509–514, https://doi.org/10.1070/ PU1968v010n04ABEH003699

    work page 1968

  23. [24]

    Ye and E

    C. Ye and E. T. Chung , Constraint energy minimizing generalized multiscale finite element method for inhomogeneous boundary value problems with high contrast coefficients , Mul- tiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 21 (2023), pp. 194–217, https://doi.org/10.1137/21m1459113

    work page doi:10.1137/21m1459113 2023

  24. [25]

    C. Ye, S. Fu, E. T. Chung, and J. Huang , A robust two-level overlapping preconditioner for darcy flow in high-contrast media , SIAM Journal on Scientific Computing, 46 (2024), pp. A3151–A3176, https://doi.org/10.1137/23M1558987

    work page doi:10.1137/23m1558987 2024

  25. [26]

    C. Ye, S. Fu, E. T. Chung, and J. Huang , A highly parallelized multiscale preconditioner for darcy flow in high-contrast media, Journal of Computational Physics, 522 (2025), p. 113603, https://doi.org/https://doi.org/10.1016/j.jcp.2024.113603

    work page doi:10.1016/j.jcp.2024.113603 2025

  26. [27]

    C. Ye, X. Jin, P. Ciarlet Jr, and E. T. Chung , Multiscale modeling for a class of high- contrast heterogeneous sign-changing problems, arXiv:2407.17130, (2024), https://doi.org/ 10.48550/arXiv.2407.17130

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2407.17130 2024

  27. [28]

    Zhao and E

    L. Zhao and E. T. Chung , An analysis of the NLMC upscaling method for high contrast problems, Journal of Computational and Applied Mathematics, 367 (2020), p. 112480, https://doi.org/10.1016/j.cam.2019.112480

    work page doi:10.1016/j.cam.2019.112480 2020

  28. [29]

    Zhao and E

    L. Zhao and E. T. Chung , Constraint Energy Minimizing Generalized Multiscale Finite El- ement Method for Convection Diffusion Equation , Multiscale Modeling & Simulation, 21 (2023), pp. 735–752, https://doi.org/https://doi.org/10.1137/22M1487655

    work page doi:10.1137/22m1487655 2023

  29. [30]

    R. W. Ziolkowski, Pulsed and cw gaussian beam interactions with double negative metama- terial slabs, Optics Express, 11 (2003), pp. 662–681. This manuscript is for review purposes only

    work page 2003