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ISRN Mathematical Physics
Volume 2013 (2013), Article ID 487270, 16 pages
http://dx.doi.org/10.1155/2013/487270
Review Article

Differential Forms in Lattice Field Theories: An Overview

ElectroScience Laboratory, Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43212, USA

Received 13 November 2012; Accepted 11 December 2012

Academic Editors: J. Banasiak, F. Sugino, and G. F. Torres del Castillo

Copyright © 2013 F. L. Teixeira. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. I. Montvay and G. Münster, Quantum Fields on a Lattice, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, UK, 1997. View at MathSciNet
  2. A. Zee, Quantum Field Theory in a Nutshell, Princeton University Press, Princeton, NJ, USA, 2003. View at Zentralblatt MATH · View at MathSciNet
  3. W. C. Chew, “Electromagnetic field theory on a lattice,” Journal of Applied Physics, vol. 75, no. 10, pp. 4843–4850, 1994. View at Publisher · View at Google Scholar
  4. L. S. Martin and Y. Oono, “Physics-motivated numerical solvers for partial differential equations,” Physical Review E, vol. 57, no. 4, pp. 4795–4810, 1998. View at Publisher · View at Google Scholar
  5. M. A. H. Lopez, S. G. Garcia, A. R. Bretones, and R. G. Martin, “Simulation of the transient response of objects buried in dispersive media,” in Ultrawideband Short-Pulse Electromagnetics, vol. 5, Kluwer Academic Press, Dordrecht, The Netherlands, 2000.
  6. F. L. Teixeira, “Time-domain finite-difference and finite-element methods for Maxwell equations in complex media,” IEEE Transactions on Antennas and Propagation, vol. 56, no. 8, part 1, pp. 2150–2166, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  7. N. H. Christ, R. Friedberg, and T. D. Lee, “Gauge theory on a random lattice,” Nuclear Physics B, vol. 210, no. 3, pp. 310–336, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  8. J. E. Bolander and N. Sukumar, “Irregular lattice model for quasistatic crack propagation,” Physical Review B, vol. 71, Article ID 094106, 2005.
  9. J. M. Drouffe and K. J. M. Moriarty, “U(2) four-dimensional simplicial lattice gauge theory,” Zeitschrift für Physik C, vol. 24, no. 3, pp. 395–403, 1984. View at Publisher · View at Google Scholar · View at Scopus
  10. M. Göckeler, “Dirac-Kähler fields and the lattice shape dependence of fermion flavour,” Zeitschrift für Physik C, vol. 18, no. 4, pp. 323–326, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  11. J. Komorowski, “On finite-dimensional approximations of the exterior differential codifferential and Laplacian on a Riemannian manifold,” Bulletin de l'Académie Polonaise des Sciences, vol. 23, no. 9, pp. 999–1005, 1975. View at Zentralblatt MATH · View at MathSciNet
  12. J. Dodziuk, “Finite-difference approach to the Hodge theory of harmonic forms,” American Journal of Mathematics, vol. 98, no. 1, pp. 79–104, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. R. Sorkin, “The electromagnetic field on a simplicial net,” Journal of Mathematical Physics, vol. 16, no. 12, pp. 2432–2440, 1975. View at Publisher · View at Google Scholar · View at MathSciNet
  14. D. Weingarten, “Geometric formulation of electrodynamics and general relativity in discrete space-time,” Journal of Mathematical Physics, vol. 18, no. 1, pp. 165–170, 1977. View at Publisher · View at Google Scholar · View at MathSciNet
  15. W. Müller, “Analytic torsion and R-torsion of Riemannian manifolds,” Advances in Mathematics, vol. 28, no. 3, pp. 233–305, 1978. View at Publisher · View at Google Scholar · View at MathSciNet
  16. P. Becher and H. Joos, “The Dirac-Kähler equation and fermions on the lattice,” Zeitschrift für Physik. C, vol. 15, no. 4, pp. 343–365, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  17. J. M. Rabin, “Homology theory of lattice fermion doubling,” Nuclear Physics. B, vol. 201, no. 2, pp. 315–332, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  18. A. Bossavit, Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements, Electromagnetism, Academic Press, San Diego, Calif, USA, 1998. View at MathSciNet
  19. A. Bossavit, “Differential forms and the computation of fields and forces in electromagnetism,” European Journal of Mechanics. B, vol. 10, no. 5, pp. 474–488, 1991. View at Zentralblatt MATH · View at MathSciNet
  20. C. Mattiussi, “An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology,” Journal of Computational Physics, vol. 133, no. 2, pp. 289–309, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. L. Kettunen, K. Forsman, and A. Bossavit, “Discrete spaces for div and curl-free fields,” IEEE Transactions on Magnetics, vol. 34, pp. 2551–2554, 1998. View at Publisher · View at Google Scholar
  22. F. L. Teixeira and W. C. Chew, “Lattice electromagnetic theory from a topological viewpoint,” Journal of Mathematical Physics, vol. 40, no. 1, pp. 169–187, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. T. Tarhasaari, L. Kettunen, and A. Bossavit, “Some realizations of a discrete Hodge operator: a reinterpretation of finite element techniques,” IEEE Transactions on Magnetics, vol. 35, no. 3, pp. 1494–1497, 1999. View at Scopus
  24. S. Sen, S. Sen, J. C. Sexton, and D. H. Adams, “Geometric discretization scheme applied to the abelian Chern-Simons theory,” Physical Review E, vol. 61, no. 3, pp. 3174–3185, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  25. J. A. Chard and V. Shapiro, “A multivector data structure for differential forms and equations,” Mathematics and Computers in Simulation, vol. 54, no. 1–3, pp. 33–64, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  26. P. W. Gross and P. R. Kotiuga, “Data structures for geometric and topological aspects of finite element algorithms,” in Geometric Methods in Computational Electromagnetics, F. L. Teixeira, Ed., vol. 32 of Progress in Electromagnetics Research, pp. 151–169, EMW Publishing, Cambridge, Mass, USA, 2001.
  27. F. L. Teixeira, “Geometrical aspects of the simplicial discretization of Maxwell’s equations,” in Geometric Methods in Computational Electromagnetics, F. L. Teixeira, Ed., vol. 32 of Progress in Electromagnetics Research, pp. 171–188, EMW Publishing, Cambridge, Mass, USA, 2001.
  28. T. Tarhasaari and L. Kettunen, “Topological approach to computational electromagnetism,” in Geometric Methods in Computational Electromagnetics, F. L. Teixeira, Ed., vol. 32 of Progress in Electromagnetics Research, pp. 189–206, EMW Publishing, Cambridge, Mass, USA, 2001.
  29. J. Kim and F. L. Teixeira, “Parallel and explicit finite-element time-domain method for Maxwell's equations,” IEEE Transactions on Antennas and Propagation, vol. 59, no. 6, part 2, pp. 2350–2356, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  30. A. S. Schwarz, Topology for Physicists, vol. 308 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1994. View at MathSciNet
  31. B. He and F. L. Teixeira, “On the degrees of freedom of lattice electrodynamics,” Physics Letters A, vol. 336, no. 1, pp. 1–7, 2005. View at Publisher · View at Google Scholar
  32. B. He and F. L. Teixeira, “Mixed E-B finite elements for solving 1-D, 2-D, and 3-D time-harmonic Maxwell curl equations,” IEEE Microwave and Wireless Components Letters, vol. 17, no. 5, pp. 313–315, 2007. View at Publisher · View at Google Scholar
  33. H. Whitney, Geometric Integration Theory, Princeton University Press, Princeton, NJ, USA, 1957. View at MathSciNet
  34. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H. Freeman, San Francisco, Calif, USA, 1973. View at MathSciNet
  35. G. A. Deschamps, “Electromagnetics and differential forms,” Proceedings of the IEEE, vol. 69, pp. 676–696, 1982.
  36. P. R. Kotiuga, “Metric dependent aspects of inverse problems and functionals based on helicity,” Journal of Applied Physics, vol. 73, no. 10, pp. 5437–5439, 1993. View at Publisher · View at Google Scholar
  37. F. L. Teixeira and W. C. Chew, “Unified analysis of perfectly matched layers using differential forms,” Microwave and Optical Technology Letters, vol. 20, no. 2, pp. 124–126, 1999. View at Publisher · View at Google Scholar
  38. F. L. Teixeira and W. C. Chew, “Differential forms, metrics, and the reflectionless absorption of electromagnetic waves,” Journal of Electromagnetic Waves and Applications, vol. 13, no. 5, pp. 665–686, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. F. L. Teixeira, “Differential form approach to the analysis of electromagnetic cloaking and masking,” Microwave and Optical Technology Letters, vol. 49, no. 8, pp. 2051–2053, 2007. View at Publisher · View at Google Scholar
  40. A. H. Guth, “Existence proof of a nonconfining phase in four-dimensional U(1) lattice gauge theory,” Physical Review D, vol. 21, no. 8, pp. 2291–2307, 1980. View at Publisher · View at Google Scholar · View at MathSciNet
  41. A. Kheyfets and W. A. Miller, “The boundary of a boundary principle in field theories and the issue of austerity of the laws of physics,” Journal of Mathematical Physics, vol. 32, no. 11, pp. 3168–3175, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  42. R. Hiptmair, “Discrete Hodge operators,” Numerische Mathematik, vol. 90, no. 2, pp. 265–289, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. B. He and F. L. Teixeira, “Geometric finite element discretization of Maxwell equations in primal and dual spaces,” Physics Letters A, vol. 349, no. 1–4, pp. 1–14, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. B. He and F. L. Teixeira, “Differential forms, Galerkin duality, and sparse inverse approximations in finite element solutions of Maxwell equations,” IEEE Transactions on Antennas and Propagation, vol. 55, no. 5, pp. 1359–1368, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  45. B. Donderici and F. L. Teixeira, “Conformal perfectly matched layer for the mixed finite element time-domain method,” IEEE Transactions on Antennas and Propagation, vol. 56, no. 4, pp. 1017–1026, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  46. W. L. Burke, Applied Differential Geometry, Cambridge University Press, Cambridge, UK, 1985. View at MathSciNet
  47. E. Tonti, “The reason for analogies between physical theories,” Applied Mathematical Modelling, vol. 1, no. 1, pp. 37–50, 1976. View at Publisher · View at Google Scholar · View at MathSciNet
  48. E. Tonti, “Finite formulation of the electromagnetic field,” in Geometric Methods in Computational Electromagnetics, F. L. Teixeira, Ed., vol. 32 of Progress in Electromagnetics Research, pp. 1–44, EMW Publishing, Cambridge, Mass, USA, 2001.
  49. E. Tonti, “On the mathematical structure of a large class of physical theories,” Rendiconti della Reale Accademia Nazionale dei Lincei, vol. 52, pp. 48–56, 1972. View at MathSciNet
  50. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’sequation is isotropic media,” IEEE Transactions on Antennas and Propagation, vol. 14, no. 3, pp. 302–307, 1969.
  51. A. Taflove, Computational Electrodynamics, Artech House, Boston, Mass, USA, 1995. View at MathSciNet
  52. R. A. Nicolaides and X. Wu, “Covolume solutions of three-dimensional div-curl equations,” SIAM Journal on Numerical Analysis, vol. 34, no. 6, pp. 2195–2203, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  53. L. Codecasa, R. Specogna, and F. Trevisan, “Symmetric positive-definite constitutive matrices for discrete eddy-current problems,” IEEE Transactions on Magnetics, vol. 43, no. 2, pp. 510–515, 2007. View at Publisher · View at Google Scholar · View at Scopus
  54. B. Auchmann and S. Kurz, “A geometrically defined discrete hodge operator on simplicial cells,” IEEE Transactions on Magnetics, vol. 42, no. 4, pp. 643–646, 2006. View at Publisher · View at Google Scholar · View at Scopus
  55. A. Bossavit, “Whitney forms: a new class of finite elements for three-dimensional computations in electromagnetics,” IEE Proceedings A, vol. 135, pp. 493–500, 1988.
  56. P. W. Gross and P. R. Kotiuga, Electromagnetic Theory and Computation: A Topological Approach, vol. 48 of Mathematical Sciences Research Institute Publications, Cambridge University Press, Cambridge, UK, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  57. A. Bossavit, “Discretization of electromagnetic problems: the “generalized finite differences” approach,” in Handbook of Numerical Analysis, vol. 13, pp. 105–197, North-Holland Publishing, Amsterdam, The Netherlands, 2005. View at Zentralblatt MATH · View at MathSciNet
  58. B. He, Compatible discretizations of Maxwell equations [Ph.D. thesis], The Ohio State University, Columbus, Ohio, USA, 2006.
  59. R. Hiptmair, “Higher order Whitney forms,” in Geometric Methods in Computational Electromagnetics, F. L. Teixeira, Ed., vol. 32 of Progress in Electromagnetics Research, pp. 271–299, EMW Publishing, Cambridge, Mass, USA, 2001.
  60. F. Rapetti and A. Bossavit, “Whitney forms of higher degree,” SIAM Journal on Numerical Analysis, vol. 47, no. 3, pp. 2369–2386, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  61. J. Kangas, T. Tarhasaari, and L. Kettunen, “Reading Whitney and finite elements with hindsight,” IEEE Transactions on Magnetics, vol. 43, no. 4, pp. 1157–1160, 2007. View at Publisher · View at Google Scholar · View at Scopus
  62. A. Buffa, J. Rivas, G. Sangalli, and R. Vázquez, “Isogeometric discrete differential forms in three dimensions,” SIAM Journal on Numerical Analysis, vol. 49, no. 2, pp. 818–844, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  63. A. Back and E. Sonnendrücker, “Spline discrete differential forms,” in Proceedings of ESAIM, vol. 35, pp. 197–202, March 2012. View at Publisher · View at Google Scholar
  64. S. Albeverio and B. Zegarliński, “Construction of convergent simplicial approximations of quantum fields on Riemannian manifolds,” Communications in Mathematical Physics, vol. 132, no. 1, pp. 39–71, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  65. S. Albeverio and J. Schäfer, “Abelian Chern-Simons theory and linking numbers via oscillatory integrals,” Journal of Mathematical Physics, vol. 36, no. 5, pp. 2157–2169, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  66. S. O. Wilson, “Cochain algebra on manifolds and convergence under refinement,” Topology and Its Applications, vol. 154, no. 9, pp. 1898–1920, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  67. S. O. Wilson, “Differential forms, fluids, and finite models,” Proceedings of the American Mathematical Society, vol. 139, no. 7, pp. 2597–2604, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  68. T. G. Halvorsen and T. M. Sørensen, “Simplicial gauge theory and quantum gauge theory simulation,” Nuclear Physics B, vol. 854, no. 1, pp. 166–183, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  69. A. Bossavit, “Computational electromagnetism and geometry: (5) the " Galerkin Hodge,” Journal of the Japan Society of Applied Electromagnetics, vol. 8, pp. 203–209, 2000.
  70. E. Katz and U. J. Wiese, “Lattice fluid dynamics from perfect discretizations of continuum flows,” Physical Review E, vol. 58, pp. 5796–5807, 1998. View at Publisher · View at Google Scholar
  71. B. He and F. L. Teixeira, “Sparse and explicit FETD via approximate inverse hodge (Mass) matrix,” IEEE Microwave and Wireless Components Letters, vol. 16, no. 6, pp. 348–350, 2006. View at Publisher · View at Google Scholar · View at Scopus
  72. D. H. Adams, “A doubled discretization of abelian Chern-Simons theory,” Physical Review Letters, vol. 78, no. 22, pp. 4155–4158, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  73. A. Buffa and S. H. Christiansen, “A dual finite element complex on the barycentric refinement,” Mathematics of Computation, vol. 76, no. 260, pp. 1743–1769, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  74. A. Gillette and C. Bajaj, “Dual formulations of mixed finite element methods with applications,” Computer-Aided Design, vol. 43, pp. 1213–1221, 2011. View at Publisher · View at Google Scholar
  75. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” Journal of Computational Physics, vol. 114, no. 2, pp. 185–200, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  76. W. C. Chew and W. H. Weedon, “3D perfectly matched medium from modified Maxwell's equations with stretched coordinates,” Microwave and Optical Technology Letters, vol. 7, no. 13, pp. 599–604, 1994. View at Scopus
  77. F. L. Teixeira and W. C. Chew, “PML-FDTD in cylindrical and spherical grids,” IEEE Microwave and Guided Wave Letters, vol. 7, no. 9, pp. 285–287, 1997. View at Scopus
  78. F. Collino and P. Monk, “The perfectly matched layer in curvilinear coordinates,” SIAM Journal on Scientific Computing, vol. 19, no. 6, pp. 2061–2090, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  79. Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “Perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Transactions on Antennas and Propagation, vol. 43, no. 12, pp. 1460–1463, 1995. View at Publisher · View at Google Scholar · View at Scopus
  80. F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave and Guided Wave Letters, vol. 7, no. 11, pp. 371–373, 1997. View at Scopus
  81. F. L. Teixeira and W. C. Chew, “Analytical derivation of a conformal perfectly matched absorber for electromagnetic waves,” Microwave and Optical Technology Letters, vol. 17, no. 4, pp. 231–236, 1998. View at Scopus
  82. B. Donderici and F. L. Teixeira, “Conformal perfectly matched layer for the mixed finite element time-domain method,” IEEE Transactions on Antennas and Propagation, vol. 56, no. 4, pp. 1017–1026, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  83. F. L. Teixeira and W. C. Chew, “On Causality and dynamic stability of perfectly matched layers for FDTD simulations,” IEEE Transactions on Microwave Theory and Techniques, vol. 47, no. 63, pp. 775–785, 1999. View at Scopus
  84. F. L. Teixeira and W. C. Chew, “Complex space approach to perfectly matched layers: a review and some new developments,” International Journal of Numerical Modelling, vol. 13, no. 5, pp. 441–455, 2000.
  85. R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, IOP Publishing, Bristol, UK, 1988.
  86. T. Z. Ezirkepov, “Exact charge conservation scheme for particle-in-cell simularion with an arbitrary form-factor,” Computer Physics Communications, vol. 135, pp. 144–153, 2001. View at Publisher · View at Google Scholar
  87. Y. A. Omelchenko and H. Karimabadi, “Event-driven, hybrid particle-in-cell simulation: a new paradigm for multi-scale plasma modeling,” Journal of Computational Physics, vol. 216, no. 1, pp. 153–178, 2006. View at Publisher · View at Google Scholar · View at Scopus
  88. P. J. Mardahl and J. P. Verboncoeur, “Charge conservation in electromagnetic PIC codes; spectral comparison of Boris/DADI and Langdon-Marder methods,” Computer Physics Communications, vol. 106, no. 3, pp. 219–229, 1997. View at Scopus
  89. F. Assous, “A three-dimensional time domain electromagnetic particle-in-cell codeon unstructured grids,” International Journal of Modelling and Simulation, vol. 29, no. 3, pp. 279–284, 2009. View at Publisher · View at Google Scholar · View at Scopus
  90. A. Candel, A. Kabel, L. Q. Lee et al., “State of the art in electromagnetic modeling for the compact linear collider,” Journal of Physics: Conference Series, vol. 180, no. 1, Article ID 012004, 2009. View at Publisher · View at Google Scholar · View at Scopus
  91. J. Squire, H. Qin, and W. M. Tang, “Geometric integration of the Vlaslov-Maxwell system with a variational partcile-in-cell scheme,” Physics of Plasmas, vol. 19, Article ID 084501, 2012.
  92. R. A. Chilton, H-, P- and T-refinement strategies for the finite-difference-time-domain (FDTD) method developed via finite-element (FE) principles [Ph.D. thesis], The Ohio State University, Columbus, Ohio, USA, 2008.
  93. K. S. Yee and J. S. Chen, “The finite-difference time-domain (FDTD) and the finite-volume time-domain (FVTD) methods in solving Maxwell's equations,” IEEE Transactions on Antennas and Propagation, vol. 45, no. 3, pp. 354–363, 1997. View at Scopus
  94. C. Mattiussi, “The geometry of time-stepping,” in Geometric Methods in Computational Electromagnetics, F. L. Teixeira, Ed., vol. 32 of Progress in Electromagnetics Research, pp. 123–149, EMW Publishing, Cambridge, Mass, USA, 2001.
  95. J. Fang, Time domain finite difference computation for Maxwell equations [Ph.D. thesis], University of California, Berkeley, Calif, USA, 1989.
  96. Z. Xie, C.-H. Chan, and B. Zhang, “An explicit fourth-order staggered finite-difference time-domain method for Maxwell's equations,” Journal of Computational and Applied Mathematics, vol. 147, no. 1, pp. 75–98, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  97. S. Wang and F. L. Teixeira, “Lattice models for large scale simulations of coherent wave scattering,” Physical Review E, vol. 69, Article ID 016701, 2004.
  98. T. Weiland, “On the numerical solution of Maxwell’s equations and applications in accelerator physics,” Particle Accelerators, vol. 15, pp. 245–291, 1996.
  99. R. Schuhmann and T. Weiland, “Rigorous analysis of trapped modes in accelerating cavities,” Physical Review Special Topics—Accelerators and Beams, vol. 3, no. 12, pp. 28–36, 2000. View at Scopus
  100. L. Codecasa, V. Minerva, and M. Politi, “Use of barycentric dual grids,” IEEE Transactions on Magnetics, vol. 40, pp. 1414–1419, 2004. View at Publisher · View at Google Scholar
  101. R. Schuhmann and T. Weiland, “Stability of the FDTD algorithm on nonorthogonal grids related to the spatial interpolation scheme,” IEEE Transactions on Magnetics, vol. 34, no. 5, pp. 2751–2754, 1998. View at Scopus
  102. R. Schuhmann, P. Schmidt, and T. Weiland, “A new Whitney-based material operator for the finite-integration technique on triangular grids,” IEEE Transactions on Magnetics, vol. 38, no. 2, pp. 409–412, 2002. View at Publisher · View at Google Scholar · View at Scopus
  103. M. Bullo, F. Dughiero, M. Guarnieri, and E. Tittonel, “Isotropic and anisotropic electrostatic field computation by means of the cell method,” IEEE Transactions on Magnetics, vol. 40, no. 2, pp. 1013–1016, 2004. View at Publisher · View at Google Scholar · View at Scopus
  104. P. Alotto, A. De Cian, and G. Molinari, “A time-domain 3-D full-Maxwell solver based on the cell method,” IEEE Transactions on Magnetics, vol. 42, no. 4, pp. 799–802, 2006. View at Publisher · View at Google Scholar · View at Scopus
  105. M. Bullo, F. Dughiero, M. Guarnieri, and E. Tittonel, “Nonlinear coupled thermo-electromagnetic problems with the cell method,” IEEE Transactions on Magnetics, vol. 42, no. 4, pp. 991–994, 2006. View at Publisher · View at Google Scholar · View at Scopus
  106. P. Alotto, M. Bullo, M. Guarnieri, and F. Moro, “A coupled thermo-electromagnetic formulation based on the cell method,” IEEE Transactions on Magnetics, vol. 44, no. 6, pp. 702–705, 2008. View at Publisher · View at Google Scholar · View at Scopus
  107. P. Alotto, F. Freschi, and M. Repetto, “Multiphysics problems via the cell method: the role of tonti diagrams,” IEEE Transactions on Magnetics, vol. 46, no. 8, pp. 2959–2962, 2010. View at Publisher · View at Google Scholar · View at Scopus
  108. L. Codecasa, R. Specogna, and F. Trevisan, “Discrete geometric formulation of admittance boundary conditions for frequency domain problems over tetrahedral dual grids,” IEEE Transactions on Antennas and Propagation, vol. 60, pp. 3998–4002, 2012. View at Publisher · View at Google Scholar
  109. M. Shashkov and S. Steinberg, “Support-operator finite-difference algorithms for general elliptic problems,” Journal of Computational Physics, vol. 118, no. 1, pp. 131–151, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  110. J. M. Hyman and M. Shashkov, “Mimetic discretizations for Maxwell's equations,” Journal of Computational Physics, vol. 151, no. 2, pp. 881–909, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  111. J. M. Hyman and M. Shashkov, “The orthogonal decomposition theorems for mimetic finite difference methods,” SIAM Journal on Numerical Analysis, vol. 36, no. 3, pp. 788–818, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  112. J. M. Hyman and M. Shashkov, “Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids,” Applied Numerical Mathematics, vol. 25, no. 4, pp. 413–442, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  113. J. M. Hyman and M. Shashkov, “Mimetic finite difference methods for Maxwell’s equations and the equations of magnetic diffusion,” in Geometric Methods in Computational Electromagnetics, F. L. Teixeira, Ed., vol. 32 of Progress in Electromagnetics Research, pp. 89–121, EMW Publishing, Cambridge, Mass, USA, 2001.
  114. J. Castillo and T. McGuinness, “Steady state diffusion problems on non-trivial domains: support operator method integrated with direct optimized grid generation,” Applied Numerical Mathematics, vol. 40, no. 1-2, pp. 207–218, 2002. View at Publisher · View at Google Scholar · View at Scopus
  115. K. Lipnikov, M. Shashkov, and D. Svyatskiy, “The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes,” Journal of Computational Physics, vol. 211, no. 2, pp. 473–491, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  116. F. Brezzi and A. Buffa, “Innovative mimetic discretizations for electromagnetic problems,” Journal of Computational and Applied Mathematics, vol. 234, no. 6, pp. 1980–1987, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  117. N. Robidoux and S. Steinberg, “A discrete vector calculus in tensor grids,” Computational Methods in Applied Mathematics, vol. 11, no. 1, pp. 23–66, 2011. View at MathSciNet
  118. K. Lipnikov, G. Manzini, F. Brezzi, and A. Buffa, “The mimetic finite difference method for the 3D magnetostatic field problems on polyhedral meshes,” Journal of Computational Physics, vol. 230, no. 2, pp. 305–328, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  119. V. Subramanian and J. B. Perot, “Higher-order mimetic methods for unstructured meshes,” Journal of Computational Physics, vol. 219, no. 1, pp. 68–85, 2006. View at Publisher · View at Google Scholar · View at Scopus
  120. L. Beirão da Veiga and G. Manzini, “A higher-order formulation of the mimetic finite difference method,” SIAM Journal on Scientific Computing, vol. 31, no. 1, pp. 732–760, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  121. D. N. Arnold, “Differential complexes and numerical stability,” in Proceedings of the International Congress of Mathematicians, vol. 1 of Plenary Lectures, pp. 137–157, Beijing, China, 2002. View at MathSciNet
  122. D. N. Arnold, P. B. Bochev, R. B. Lehoucq, R. A. Nicolaides, and M. Shashkov, Eds., Compatible Spatial Discretizations, vol. 142 of The IMA Volumes in Mathematics and Its Applications, Springer, New York, NY, USA, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  123. D. A. White, J. M. Koning, and R. N. Rieben, “Development and application of compatible discretizations of Maxwell's equations,” in Compatible Spatial Discretizations, vol. 142 of The IMA Volumes in Mathematics and Its Applications, pp. 209–234, Springer, New York, NY, USA, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  124. P. Bochev and M. Gunzburger, “Compatible discretizations of second-order elliptic problems,” Journal of Mathematical Sciences, vol. 136, no. 2, pp. 3691–3705, 2006. View at Publisher · View at Google Scholar
  125. D. Boffi, “Approximation of eigenvalues in mixed form, discrete compactness property, and application to hp mixed finite elements,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 37–40, pp. 3672–3681, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  126. P. Bochev, H. C. Edwards, R. C. Kirby, K. Peterson, and D. Ridzal, “Solving PDEs with Intrepid,” Scientific Programming, vol. 20, pp. 151–180, 2012.
  127. J.-C. Nédélec, “Mixed finite elements in 𝐑3,” Numerische Mathematik, vol. 35, no. 3, pp. 315–341, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  128. R. Hiptmair, “Canonical construction of finite elements,” Mathematics of Computation, vol. 68, no. 228, pp. 1325–1346, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  129. V. W. Guillemin and S. Sternberg, Supersymmetry and Equivariant de Rham Theory, Mathematics Past and Present, Springer, Berlin, Germany, 1999. View at MathSciNet
  130. D. N. Arnold, R. S. Falk, and R. Winther, “Finite element exterior calculus, homological techniques, and applications,” Acta Numerica, vol. 15, pp. 1–155, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  131. A. Yavari, “On geometric discretization of elasticity,” Journal of Mathematical Physics, vol. 49, no. 2, article 022901, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  132. A. Bossavit, “Mixed finite elements and the complex of Whitney forms,” in The Mathematics of Finite Elements and Applications, VI (Uxbridge, 1987), J. R. Whiteman, Ed., pp. 137–144, Academic Press, London, UK, 1988. View at MathSciNet
  133. M. F. Wong, O. Picon, and V. F. Hanna, “Finite element method based on Whitney forms to solve Maxwell equations in the time domain,” IEEE Transactions on Magnetics, vol. 31, no. 3, pp. 1618–1621, 1995. View at Scopus
  134. M. Feliziani and F. Maradei, “Mixed finite-difference/Whitney-elements time-domain (FD/WE-TD) method,” IEEE Transactions on Magnetics, vol. 34, no. 5, pp. 3222–3227, 1998. View at Publisher · View at Google Scholar
  135. P. Castillo, J. Koning, R. Rieben, and D. White, “A discrete differential forms framework for computational electromagnetism,” Computer Modeling in Engineering & Sciences, vol. 5, no. 4, pp. 331–345, 2004. View at Zentralblatt MATH · View at MathSciNet
  136. R. N. Rieben, G. H. Rodrigue, and D. A. White, “A high order mixed vector finite element method for solving the time dependent Maxwell equations on unstructured grids,” Journal of Computational Physics, vol. 204, no. 2, pp. 490–519, 2005. View at Publisher · View at Google Scholar · View at Scopus
  137. M. Dsebrun, A. N. Hirani, and J. E. Mardsen, “Discrete exterior calculus for variational problem in computer vision and graphics,” in Proceedings of the 42nd IEEE Conference on Decision and Control, pp. 4902–4907, Maui, Hawaii, USA, 2003.
  138. A. N. Hirani, Discrete exterior calculus [Ph.D. thesis], California Institute of Technology, Pasadena, Calif, USA, 2003.
  139. M. Desbrun, A. N. Hirani, M. Leok, and J. E. Mardsen, “Discrete exterior calculus,” 2005, http://arxiv.org/abs/math/0508341.
  140. A. Gillette, “Notes on discrete exterior calculus,” Tech. Rep., University of Texas at Austin, Austin, Texas, USA, 2009.
  141. J. B. Perot, “Discrete conservation properties of unstructures mesh schemes,” Annual Review of Fluid Mechanics, vol. 43, pp. 299–318, 2011. View at Publisher · View at Google Scholar
  142. P. R. Kotiuga, “Theoretical limitation of discrete exterior calculus in the context of computational electromagnetics,” IEEE Transactions on Magnetics, vol. 44, pp. 1162–1165, 2008. View at Publisher · View at Google Scholar
  143. W. Graf, “Differential forms as spinors,” Annales de l'Institut Henri Poincaré A. Physique Théorique, vol. 29, no. 1, pp. 85–109, 1978. View at Zentralblatt MATH · View at MathSciNet
  144. D. H. Adams, “Fourth root prescription for dynamical staggered fermions,” Physical Review D, vol. 72, Article ID 114512, 2005.
  145. D. Friedan, “A proof of the Nielsen-Ninomiya theorem,” Communications in Mathematical Physics, vol. 85, no. 4, pp. 481–490, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  146. I. F. Herbut, “Time reversal, fermion doubling, and the masses of lattice Dirac fermions in three dimensions,” Physical Review B, vol. 83, no. 24, Article ID 245445, 2011. View at Publisher · View at Google Scholar · View at Scopus
  147. H. Raszillier, “Lattice degeneracies of fermions,” Journal of Mathematical Physics, vol. 25, no. 6, pp. 1682–1693, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  148. I. Kanamori and N. Kawamoto, “Dirac-Kähler femion with noncommutative differential forms on a lattice,” Nuclear Physics B—Proceedings Supplements, vol. 129, pp. 877–879, 2004. View at Publisher · View at Google Scholar
  149. L. Susskind, “Lattice fermions,” Physical Review D, vol. 16, no. 10, pp. 3031–3039, 1977. View at Publisher · View at Google Scholar
  150. M. G. do Amaral, M. Kischinhevsky, C. A. A. de Carvalho, and F. L. Teixeira, “An efficient method to calculate field theories with dynamical fermions,” International Journal of Modern Physics C, vol. 2, no. 2, pp. 561–600, 1991. View at Publisher · View at Google Scholar
  151. I. M. Benn and R. W. Tucker, “The Dirac equation in exterior form,” Communications in Mathematical Physics, vol. 98, no. 1, pp. 53–63, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  152. V. de Beauce, S. Sen, and J. C. Sexton, “Chiral dirac fermions on the latice using geometric discretization,” Nuclear Physics B—Proceedings Supplements, vol. 129, pp. 468–470, 2004. View at Publisher · View at Google Scholar
  153. D. H. Adams, “Theoretical foundation for the index theorem on the lattice with staggered fermions,” Physical Review Letters, vol. 104, Article ID 141602, 2010.
  154. F. Fillion-Gourdeau, E. Lorin, and A. D. Bandrauk, “Numerical solution of the time-dependent Dirac equation in coordinate space without fermion-doubling,” Computer Physics Communications, vol. 183, no. 7, pp. 1403–1415, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  155. D. N. Arnold, “Differential complexes and numerical stability,” in Proceedings of the International Congress of Mathematicians, vol. 1 of Plenary Lecture, pp. 137–157, Higher Ed. Press, Beijing, China, 2002. View at MathSciNet
  156. J. R. Munkres, Topology, Pearson, 2nd edition, 2000.
  157. M. W. Chevalier, R. J. Luebbers, and V. P. Cable, “FDTD local grid with material traverse,” IEEE Transactions on Antennas and Propagation, vol. 45, pp. 411–421, 1997.
  158. M. J. White, Z. Yun, and M. F. Iskander, “A new 3-D FDTD multigrid technique with dielectric traverse capabilities,” IEEE Transactions on Microwave Theory and Techniques, vol. 49, no. 3, pp. 422–430, 2001. View at Publisher · View at Google Scholar · View at Scopus
  159. S. H. Christiansen and T. G. Halvorsen, “A simplicial gauge theory,” Journal of Mathematical Physics, vol. 53, no. 3, Article ID 033501, 17 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  160. A. Bossavit, “Generalized finite differences’in computational electromagnetics,” in Geometric Methods for Computational Electromagnetics, F. L. Teixeira, Ed., vol. 32 of Progress in Electromagnetics Research, pp. 45–64, EMW Publishing, Cambridge, Mass, USA, 2001.
  161. P. Thoma and T. Weiland, “A consistent subgridding scheme for the finite difference time domain method,” International Journal of Numerical Modelling, vol. 9, no. 5, pp. 359–374, 1996. View at Scopus
  162. K. M. Krishnaiah and C. J. Railton, “Passive equivalent circuit of FDTD: an application to subgridding,” Electronics Letters, vol. 33, no. 15, pp. 1277–1278, 1997. View at Scopus