Open Access
Issue
ESAIM: M2AN
Volume 58, Number 6, November-December 2024
Special issue - To commemorate Assyr Abdulle
Page(s) 2187 - 2223
DOI https://doi.org/10.1051/m2an/2024013
Published online 04 December 2024
  1. A. Abdulle, M.E. Huber and S. Lemaire, An optimization-based numerical method for diffusion problems with sign-changing coefficients. C. R. Math. 355 (2017) 472–478. [CrossRef] [MathSciNet] [Google Scholar]
  2. Y.A. Abramovich and C.D. Aliprantis, An invitation to operator theory. In Vol. 50 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2002). [Google Scholar]
  3. C. Bernardi, M. Dauge and Y. Maday, Compatibilité de traces aux arätes et coins d’un polyèdre. C. R. Acad. Sci. Paris Ser. I 331 (2000) 679–684. [CrossRef] [MathSciNet] [Google Scholar]
  4. C. Bernardi, Y. Maday and F. Rapetti, Discrétisations variationnelles de problèmes aux limites elliptiques. In Vol. 45 of Mathématiques & Applications. Springer-Verlag, Berlin (2004). [Google Scholar]
  5. A.-S. Bonnet-Ben Dhia, L. Chesnel and P. Ciarlet Jr., T-coercivity for scalar interface problems between dielectrics and metamaterials. ESAIM:M2AN 46 (2012) 1363–1387. [CrossRef] [EDP Sciences] [Google Scholar]
  6. A.-S. Bonnet-Ben Dhia, L. Chesnel and X. Claeys, Radiation condition for a non-smooth interface between a dielectric and a metamaterial. Math. Models Methods Appl. Sci. 23 (2013) 1629–1662. [CrossRef] [MathSciNet] [Google Scholar]
  7. A.-S. Bonnet-Ben Dhia, P. Ciarlet Jr. and C.-M. Zwölf, Time harmonic wave diffraction problems in materials with sign-shifting coefficients. J. Comput. Appl. Math. 234 (2010) 1912–1919. [CrossRef] [MathSciNet] [Google Scholar]
  8. A.-S. Bonnet-Ben Dhia, C. Carvalho and P. Ciarlet Jr., Mesh requirements for the finite element approximation of problems with sign-changing coefficients. Numer. Math. 138 (2018) 801–838. [CrossRef] [MathSciNet] [Google Scholar]
  9. G. Bouchitté and D. Felbacq, Homogenization near resonances and artificial magnetism from dielectrics. C. R. Acad. Sci. Paris Ser. I 339 (2004) 377–382. [CrossRef] [Google Scholar]
  10. G. Bouchitté and B. Schweizer, Cloaking of small objects by anomalous localized resonance. Quart. J. Mech. Appl. Math. 63 (2010) 437–463. [CrossRef] [MathSciNet] [Google Scholar]
  11. G. Bouchitté and B. Schweizer, Homogenization of Maxwell’s equations in a split ring geometry. SIAM Multiscale Model. Simul. 8 (2010) 717–750. [CrossRef] [MathSciNet] [Google Scholar]
  12. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, 3rd edition. In Vol. 15 of Texts in Applied Mathematics. Springer, New York (2008). [Google Scholar]
  13. R. Bunoiu and K. Ramdani, Homogenization of materials with sign changing coefficients. Commun. Math. Sci. 14 (2016) 1137–1154. [CrossRef] [Google Scholar]
  14. E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations. SIAM J. Sci. Comput. 35 (2013) A2752–A2780. [CrossRef] [Google Scholar]
  15. E. Burman, Stabilised finite element methods for ill-posed problems with conditional stability, edited by G.R. Barrenechea, F. Brezzi, A. Cangiani and E.H. Georgoulis, Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. In Vol. 114 of Lect. Notes Comput. Sci. Eng. Springer (2016) 93–127. [Google Scholar]
  16. C. Carvalho, L. Chesnel and P. Ciarlet Jr., Eigenvalue problems with sign-changing coefficients. C. R. Math. 355 (2017) 671–675. [CrossRef] [MathSciNet] [Google Scholar]
  17. C. Carvalho, P. Ciarlet Jr. and C. Scheid, Limiting amplitude principle and resonances in plasmonic structures with corners: numerical investigation. Comput. Methods Appl. Mech. Eng. 388 (2022) 23. [Google Scholar]
  18. M. Cassier, C. Hazard and P. Joly, Spectral theory for Maxwell’s equations at the interface of a metamaterial. Part I: Generalized Fourier transform. Commun. Partial Differ. Equ. 42 (2017) 1707–1748. [CrossRef] [Google Scholar]
  19. M. Cassier, C. Hazard and P. Joly, Spectral theory for Maxwell’s equations at the interface of a metamaterial. Part II: Limiting absorption, limiting amplitude principles and interface resonance. Commun. Partial Differ. Equ. 47 (2022) 1217–1295. [CrossRef] [Google Scholar]
  20. J. Cheng and M. Yamamoto, One new strategy for a priori choice of regularizing parameters in Tikhonov’s regularization. Inverse Probl. 16 (2000) L31–L38. [CrossRef] [Google Scholar]
  21. L. Chesnel and P. Ciarlet Jr., T-coercivity and continuous Galerkin methods: application to transmission problems with sign-changing coefficients. Numer. Math. 124 (2013) 1–29. [CrossRef] [MathSciNet] [Google Scholar]
  22. P.G. Ciarlet, The finite element method for elliptic problems. In Vol. 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). [Google Scholar]
  23. P. Ciarlet Jr., T-coercivity: application to the discretization of Helmholtz-like problems. Comput. Math. Appl. 64 (2012) 22–34. [CrossRef] [MathSciNet] [Google Scholar]
  24. P. Ciarlet Jr., D. Lassounon and M. Rihani, An optimal control-based numerical method for scalar transmission problems with sign-changing coefficients. SIAM J. Numer. Anal. 61 (2023) 1316–1339. [CrossRef] [MathSciNet] [Google Scholar]
  25. P.-H. Cocquet, P.-A. Mazet and V. Mouysset, On the existence and uniqueness of a solution for some frequency-dependent partial differential equations coming from the modeling of metamaterials. SIAM J. Math. Anal. 44 (2012) 3806–3833. [CrossRef] [MathSciNet] [Google Scholar]
  26. M. Costabel and E. Stephan, A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106 (1985) 367–413. [CrossRef] [MathSciNet] [Google Scholar]
  27. A. Ern and J.-L. Guermond, Finite element quasi-interpolation and best approximation. ESAIM:M2AN 51 (2017) 1367–1385. [CrossRef] [EDP Sciences] [Google Scholar]
  28. A. Ern and J.-L. Guermond, Finite Elements I: Approximation and interpolation. In Vol. 72 of Texts in Applied Mathematics. Springer, Cham (2021). [Google Scholar]
  29. A. Ern and J.-L. Guermond, Finite Elements II: Galerkin approximation, elliptic and mixed PDEs. In Vol. 73 of Texts in Applied Mathematics. Springer, Cham (2021). [Google Scholar]
  30. P. Fernandes and M. Raffetto, Well-posedness and finite element approximability of time-harmonic electromagnetic boundary value problems involving bianisotropic materials and metamaterials. Math. Models Methods Appl. Sci. 19 (2009) 2299–2335. [CrossRef] [MathSciNet] [Google Scholar]
  31. P. Grisvard, Elliptic problems in nonsmooth domains. In Vol. 24 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, MA (1985). [Google Scholar]
  32. M.D. Gunzburger, J.S. Peterson and H.K. Lee, An optimization-based domain decomposition method for partial differential equations. Comput. Math. Appl. 37 (1999) 77–93. [CrossRef] [MathSciNet] [Google Scholar]
  33. M.D. Gunzburger, M. Heinkenschloss and H.K. Lee, Solution of elliptic partial differential equations by an optimization-based domain decomposition method. Appl. Math. Comput. 113 (2000) 111–139. [MathSciNet] [Google Scholar]
  34. M. Halla, On the approximation of dispersive electromagnetic eigenvalue problems in two dimensions. IMA J. Numer. Anal. 43 (2023) 535–559. [CrossRef] [MathSciNet] [Google Scholar]
  35. C. Hazard and S. Paolantoni, Spectral analysis of polygonal cavities containing a negative-index material. Ann. H. Lebesgue 3 (2020) 1161–1193. [CrossRef] [MathSciNet] [Google Scholar]
  36. Y. Lai, H. Chen, Z.-Q. Zhang and C.T. Chan, Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell. Phys. Rev. Lett. 102 (2009) 093901. [CrossRef] [PubMed] [Google Scholar]
  37. A. Lamacz and B. Schweizer, A negative-index meta-material for Maxwell’s equations. SIAM J. Math. Anal. 48 (2016) 4155–4174. [CrossRef] [MathSciNet] [Google Scholar]
  38. J. Li, A literature survey of mathematical study of metamaterials. Int. J. Numer. Anal. Model. 13 (2016) 230–243. [MathSciNet] [Google Scholar]
  39. M.W. Licht, Smoothed projections and mixed boundary conditions. Math. Comput. 88 (2019) 607–635. [Google Scholar]
  40. W. McLean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000). [Google Scholar]
  41. G.W. Milton and N.-A. Nicorovici, On the cloaking effects associated with anomalous localized resonance. Proc. R. Soc. Lond. Ser. A 462 (2006) 3027–3059. [Google Scholar]
  42. H.-M. Nguyen, Asymptotic behavior of solutions to the Helmholtz equations with sign-changing coefficients. Trans. Amer. Math. Soc. 367 (2015) 6581–6595. [Google Scholar]
  43. H.-M. Nguyen, Cloaking via anomalous localized resonance for doubly complementary media in the quasistatic regime. J. Eur. Math. Soc. 17 (2015) 1327–1365. [CrossRef] [MathSciNet] [Google Scholar]
  44. H.-M. Nguyen, Superlensing using complementary media. Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015) 471–484. [CrossRef] [MathSciNet] [Google Scholar]
  45. H.-M. Nguyen, Cloaking using complementary media in the quasistatic regime. Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016) 1509–1518. [CrossRef] [MathSciNet] [Google Scholar]
  46. H.-M. Nguyen, Limiting absorption principle and well-posedness for the Helmholtz equation with sign-changing coefficients. J. Math. Pures Appl. 106 (2016) 342–374. [CrossRef] [MathSciNet] [Google Scholar]
  47. H.-M. Nguyen, Negative index materials: some mathematical perspectives. Acta Math. Vietnam. 44 (2019) 325–349. [CrossRef] [MathSciNet] [Google Scholar]
  48. H.-M. Nguyen and V. Vinoles, Electromagnetic wave propagation in media consisting of dispersive metamaterials. C. R. Math. 356 (2018) 757–775. [CrossRef] [MathSciNet] [Google Scholar]
  49. H.-M. Nguyen and S. Sil, Limiting absorption principle and well-posedness for the time-harmonic Maxwell equations with anisotropic sign-changing coefficients. Comm. Math. Phys. 379 (2020) 145–176. [CrossRef] [MathSciNet] [Google Scholar]
  50. S. Nicaise and J. Venel, A posteriori error estimates for a finite element approximation of transmission problems with sign-changing coefficients. J. Comput. Appl. Math. 235 (2011) 4272–4282. [CrossRef] [MathSciNet] [Google Scholar]
  51. N.-A. Nicorovici, R. C. McPhedran and G. W. Milton, Optical and dielectric properties of partially resonant composites. Phys. Rev. B 49 (1994) 8479–8482. [CrossRef] [PubMed] [Google Scholar]
  52. P. Ola, Remarks on a transmission problem. J. Math. Anal. Appl. 196 (1995) 639–658. [CrossRef] [MathSciNet] [Google Scholar]
  53. J.B. Pendry, Negative refraction makes a perfect lens. Phys. Rev. Lett. 85 (2000) 3966–3969. [CrossRef] [PubMed] [Google Scholar]
  54. A.H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comput. 28 (1974) 959–962. [CrossRef] [Google Scholar]
  55. R.A. Shelby, D.R. Smith and S. Schultz, Experimental verification of a negative index of refraction. Science 292 (2001) 77–79. [CrossRef] [Google Scholar]
  56. D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser and S. Schultz, Composite medium with simultaneously negative permeability and permittivity. Phys. Rev. Lett. 84 (2000) 4184–4187. [CrossRef] [PubMed] [Google Scholar]
  57. A.N. Tikhonov, On the solution of ill-posed problems and the method of regularization. Dokl. Akad. Nauk SSSR 151 (1963) 501–504. [Google Scholar]
  58. V.G. Veselago, The electrodynamics of substances with simultaneously negative values of ɛ and μ. Sov. Phys. Usp. 10 (1968) 509–514. [CrossRef] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you