EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 56 / No 3 (May-June 2022) ESAIM: M2AN, 56 3 (2022) 1027-1051 References
Open Access
Issue
ESAIM: M2AN
Volume 56, Number 3, May-June 2022
Page(s) 1027 - 1051
DOI https://doi.org/10.1051/m2an/2022027
Published online 13 May 2022
  1. I. Babuška and J. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. Math. Comput. 52 (1989) 275–297. [CrossRef] [Google Scholar]
  2. I. Babuška and J. Osborn, Eigenvalue Problems. Elsevier (1991) 641–787. [Google Scholar]
  3. R. Beck, R. Hiptmair, R. Hoppe and B. Wohlmuth, Residual based a posteriori error estimators for eddy current computation. ESAIM: M2AN 34 (2000) 159–182. [CrossRef] [EDP Sciences] [Google Scholar]
  4. D. Boffi, L. Gastaldi, R. Rodríguez and I. Šebestová, Residual-based a posteriori error estimation for the Maxwell’s eigenvalue problem. IMA J. Numer. Anal. 37 (2017) 1710–1732. [MathSciNet] [Google Scholar]
  5. D. Boffi, L. Gastaldi, R. Rodríguez and I. Šebestová, A posteriori error estimates for Maxwell’s eigenvalue problem. J. Sci. Comput. 78 (2019) 1250–1271. [CrossRef] [MathSciNet] [Google Scholar]
  6. S.C. Brenner, J. Sun and L. Sung, Hodge decomposition methods for a quad-curl problem on planar domains. J. Comput. Sci. 73 (2017) 495–513. [CrossRef] [Google Scholar]
  7. S.C. Brenner, J. Cui and L. Sung, Multigrid methods based on Hodge decomposition for a quad-curl problem. Comput. Methods Appl. Math. 19 (2019) 215–232. [CrossRef] [MathSciNet] [Google Scholar]
  8. F. Cakoni and H. Haddar, A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media. Inverse Prob. Imaging 1 (2017) 443–456. [Google Scholar]
  9. F. Cakoni, D. Colton, P. Monk and J. Sun, The inverse electromagnetic scattering problem for anisotropic media. Inverse Prob. 26 (2010). [Google Scholar]
  10. S. Cao, L. Chen and X. Huang, Error analysis of a decoupled finite element method for quad-curl problems. J. Sci. Comput. 90 (2022) 1–25. [CrossRef] [Google Scholar]
  11. L. Chen, Sobolev Spaces and Elliptic Equations. Course Notes (2016). [Google Scholar]
  12. H. Chen, J. Li, W. Qiu and C. Wang, A mixed finite element scheme for quad-curl source and eigenvalue problems. Commun. Comput. Phys. 29 (2021) 1125–1151. [CrossRef] [MathSciNet] [Google Scholar]
  13. G. Chen, W. Qiu and L. Xu, Analysis of an interior penalty DG method for the quad-curl problem. IMA J. Numer. Anal. 41 (2021) 2990–3023. [CrossRef] [MathSciNet] [Google Scholar]
  14. S. Christiansen and K. Hu, Generalized finite element systems for smooth differential forms and stokes’ problem. Numer. Math. 140 (2018) 327–371. [CrossRef] [MathSciNet] [Google Scholar]
  15. S. Cochez-Dhondt and S. Nicaise, Robust a posteriori error estimation for the Maxwell equations. Comput. Methods Appl. Mech. Eng. 196 (2007) 2583–2595. [CrossRef] [Google Scholar]
  16. M. Costabel and A. McIntosh, On bogovski and regularized poincaré integral operators for de rham complexes on lipschitz domains. Math. Z. 265 (2010) 297–320. [CrossRef] [MathSciNet] [Google Scholar]
  17. X. Dai, J. Xu and A. Zhou, Convergence and optimal complexity of adaptive finite element eigenvalue computations. Numer. Math. 110 (2008) 313–355. [Google Scholar]
  18. M. Dauge, Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions. Vol 1341. Springer (2006). [Google Scholar]
  19. A.-S.B.-B. Dhia, C. Hazard and S. Lohrengel, A singular field method for the solution of Maxwell’s equations in polyhedral domains. SIAM J. Appl. Math. 59 (1999) 2028–2044. [CrossRef] [MathSciNet] [Google Scholar]
  20. V. Girault and P. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Vol 5. Springer Science & Business Media (2012). [Google Scholar]
  21. J. Han, Shifted inverse iteration based multigrid methods for the quad-curl eigenvalue problem. Appl. Math. Comput. 367 (2020) 124770. [MathSciNet] [Google Scholar]
  22. Q. Hong, J. Hu, S. Shu and J. Xu, A discontinuous Galerkin method for the fourth-order curl problem. J. Comput. Math. 30 (2012) 565–578. [CrossRef] [MathSciNet] [Google Scholar]
  23. K. Hu, Q. Zhang and Z. Zhang, Simple curl-curl-conforming finite elements in two dimensions. SIAM J. Sci. Comput. 42 (2020) A3859–A3877. [CrossRef] [Google Scholar]
  24. K. Hu, Q. Zhang and Z. Zhang, A family of finite element stokes complexes in three dimensions. SIAM J. Numer. Anal. 60 (2022) 222–243. [CrossRef] [MathSciNet] [Google Scholar]
  25. P. Monk, A posteriori error indicators for Maxwell’s equations. J. Comput. Appl. Math. 100 (1998) 173–190. [CrossRef] [MathSciNet] [Google Scholar]
  26. P. Monk, Finite Element Methods for Maxwell’s Equations. Oxford University Press (2003). [Google Scholar]
  27. P. Monk and J. Sun, Finite element methods for Maxwell’s transmission eigenvalues. SIAM J. Sci. Comput. 34 (2012) B247–B264. [CrossRef] [Google Scholar]
  28. S. Nicaise, Singularities of the quad-curl problem. J. Differ. Equ. 264 (2018) 5025–5069. [CrossRef] [Google Scholar]
  29. J. Osborn, Spectral approximation for compact operators. Math. Comput. 29 (1975) 712–725. [CrossRef] [MathSciNet] [Google Scholar]
  30. J. Schöberl, A posteriori error estimates for Maxwell equations. Math. Comput. 77 (2008) 633–649. [Google Scholar]
  31. L. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. [CrossRef] [Google Scholar]
  32. J. Sun, Iterative methods for transmission eigenvalues. SIAM J. Numer. Anal. 49 (2011) 1860–1874. [CrossRef] [MathSciNet] [Google Scholar]
  33. J. Sun, A mixed FEM for the quad-curl eigenvalue problem. Numer. Math. 132 (2016) 185–200. [CrossRef] [MathSciNet] [Google Scholar]
  34. J. Sun and A. Zhou, Finite Element Methods for Eigenvalue Problems. Chapman and Hall/CRC, Boca Raton, FL (2016). [CrossRef] [Google Scholar]
  35. Z. Sun, J. Cui, F. Gao and C. Wang, Multigrid methods for a quad-curl problem based on C0 interior penalty method. Comput. Math. App. 76 (2018) 2192–2211. [Google Scholar]
  36. J. Sun, Q. Zhang and Z. Zhang, A curl-conforming weak Galerkin method for the quad-curl problem. BIT Numer. Math. 59 (2019) 1093–1114. [CrossRef] [Google Scholar]
  37. R. Verfürth, A review of a posteriori error estimation techniques for elasticity problems. Comput. Methods Appl. Mech. Eng. 176 (1999) 419–440. [Google Scholar]
  38. C. Wang, Z. Sun and J. Cui, A new error analysis of a mixed finite element method for the quad-curl problem. Appl. Math. Comput. 349 (2019) 23–38. [MathSciNet] [Google Scholar]
  39. L. Wang, H. Li and Z. Zhang, H(curl2)-conforming spectral element method for quad-curl problems. Comput. Methods Appl. Math. 21 (2021) 661–681. [CrossRef] [MathSciNet] [Google Scholar]
  40. L. Wang, W. Shan, H. Li and Z. Zhang, H(curl2)-conforming quadrilateral spectral element method for quad-curl problems. Math. Models Methods Appl. Sci. 31 (2021) 1951–1986. [CrossRef] [MathSciNet] [Google Scholar]
  41. S. Zhang, Mixed schemes for quad-curl equations. ESAIM: M2AN 52 (2018) 147–161. [CrossRef] [EDP Sciences] [Google Scholar]
  42. S. Zhang, Regular decomposition and a framework of order reduced methods for fourth order problems. Numer. Math. 138 (2018) 241–271. [CrossRef] [MathSciNet] [Google Scholar]
  43. Q. Zhang, New conforming finite elements based on the de rham complexes for some fourth-order problems. Ph.D. dissertation (2021). [Google Scholar]
  44. Q. Zhang, L. Wang and Z. Zhang, H(curl2)-conforming finite elements in 2 dimensions and applications to the quad-curl problem. SIAM J. Sci. Comput. 41 (2019) A1527–A1547. [CrossRef] [Google Scholar]
  45. B. Zheng and J. Xu, A nonconforming finite element method for fourth order curl equations in ℝ3. Math. Comput. 80 (2011) 1871–1886. [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