Vol. 32
Latest Volume
All Volumes
PIER 179 [2024] PIER 178 [2023] PIER 177 [2023] PIER 176 [2023] PIER 175 [2022] PIER 174 [2022] PIER 173 [2022] PIER 172 [2021] PIER 171 [2021] PIER 170 [2021] PIER 169 [2020] PIER 168 [2020] PIER 167 [2020] PIER 166 [2019] PIER 165 [2019] PIER 164 [2019] PIER 163 [2018] PIER 162 [2018] PIER 161 [2018] PIER 160 [2017] PIER 159 [2017] PIER 158 [2017] PIER 157 [2016] PIER 156 [2016] PIER 155 [2016] PIER 154 [2015] PIER 153 [2015] PIER 152 [2015] PIER 151 [2015] PIER 150 [2015] PIER 149 [2014] PIER 148 [2014] PIER 147 [2014] PIER 146 [2014] PIER 145 [2014] PIER 144 [2014] PIER 143 [2013] PIER 142 [2013] PIER 141 [2013] PIER 140 [2013] PIER 139 [2013] PIER 138 [2013] PIER 137 [2013] PIER 136 [2013] PIER 135 [2013] PIER 134 [2013] PIER 133 [2013] PIER 132 [2012] PIER 131 [2012] PIER 130 [2012] PIER 129 [2012] PIER 128 [2012] PIER 127 [2012] PIER 126 [2012] PIER 125 [2012] PIER 124 [2012] PIER 123 [2012] PIER 122 [2012] PIER 121 [2011] PIER 120 [2011] PIER 119 [2011] PIER 118 [2011] PIER 117 [2011] PIER 116 [2011] PIER 115 [2011] PIER 114 [2011] PIER 113 [2011] PIER 112 [2011] PIER 111 [2011] PIER 110 [2010] PIER 109 [2010] PIER 108 [2010] PIER 107 [2010] PIER 106 [2010] PIER 105 [2010] PIER 104 [2010] PIER 103 [2010] PIER 102 [2010] PIER 101 [2010] PIER 100 [2010] PIER 99 [2009] PIER 98 [2009] PIER 97 [2009] PIER 96 [2009] PIER 95 [2009] PIER 94 [2009] PIER 93 [2009] PIER 92 [2009] PIER 91 [2009] PIER 90 [2009] PIER 89 [2009] PIER 88 [2008] PIER 87 [2008] PIER 86 [2008] PIER 85 [2008] PIER 84 [2008] PIER 83 [2008] PIER 82 [2008] PIER 81 [2008] PIER 80 [2008] PIER 79 [2008] PIER 78 [2008] PIER 77 [2007] PIER 76 [2007] PIER 75 [2007] PIER 74 [2007] PIER 73 [2007] PIER 72 [2007] PIER 71 [2007] PIER 70 [2007] PIER 69 [2007] PIER 68 [2007] PIER 67 [2007] PIER 66 [2006] PIER 65 [2006] PIER 64 [2006] PIER 63 [2006] PIER 62 [2006] PIER 61 [2006] PIER 60 [2006] PIER 59 [2006] PIER 58 [2006] PIER 57 [2006] PIER 56 [2006] PIER 55 [2005] PIER 54 [2005] PIER 53 [2005] PIER 52 [2005] PIER 51 [2005] PIER 50 [2005] PIER 49 [2004] PIER 48 [2004] PIER 47 [2004] PIER 46 [2004] PIER 45 [2004] PIER 44 [2004] PIER 43 [2003] PIER 42 [2003] PIER 41 [2003] PIER 40 [2003] PIER 39 [2003] PIER 38 [2002] PIER 37 [2002] PIER 36 [2002] PIER 35 [2002] PIER 34 [2001] PIER 33 [2001] PIER 32 [2001] PIER 31 [2001] PIER 30 [2001] PIER 29 [2000] PIER 28 [2000] PIER 27 [2000] PIER 26 [2000] PIER 25 [2000] PIER 24 [1999] PIER 23 [1999] PIER 22 [1999] PIER 21 [1999] PIER 20 [1998] PIER 19 [1998] PIER 18 [1998] PIER 17 [1997] PIER 16 [1997] PIER 15 [1997] PIER 14 [1996] PIER 13 [1996] PIER 12 [1996] PIER 11 [1995] PIER 10 [1995] PIER 09 [1994] PIER 08 [1994] PIER 07 [1993] PIER 06 [1992] PIER 05 [1991] PIER 04 [1991] PIER 03 [1990] PIER 02 [1990] PIER 01 [1989]
0000-00-00
Discrete Hodge-Operators: an Algebraic Perspective
By
, Vol. 32, 247-269, 2001
Abstract
Discrete differential forms should be used to deal with the discretization of boundary value problems that can be stated in the calculus of differential forms. This approach preserves the topological features of the equations. Yet, the discrete counterparts of the metricdependent constitutive laws remain elusive. I introduce a few purely algebraic constraints that matrices associated with discrete material laws have to satisfy. It turns out that most finite element and finite volume schemes comply with these requirements. Thus convergence analysis can be conducted in a unified setting. This discloses basic sufficient conditions that discrete material laws have to meet in order to ensure convergence in the relevant energy norms.
Citation
R. Hiptmair, "Discrete Hodge-Operators: an Algebraic Perspective," , Vol. 32, 247-269, 2001.
doi:10.2528/PIER00080110
References

1. Bank, R. and D. Rose, "Some error estimates for the box method," SIAM J. Numer. Anal., Vol. 24, 777-787, 1987.
doi:10.1137/0724050

2. Baranger, J., J.-F. Maitre, and F. Oudin, "Connection between finite volume and mixed finite element methods," RAIRO, Modelisation Math. Anal. Numer., Vol. 30, 445-465, 1996.
doi:10.1051/m2an/1996300404451

3. Bossavit, A., "Mixed finite elements and the complex of Whitney forms," The Mathematics of Finite Elements and Applications VI, J. Whiteman (ed.), 137–144, Academic Press, London, 1988.

4. Bossavit, A., "A new viewpoint on mixed elements," Meccanica, Vol. 27, 3-11, 1992.
doi:10.1007/BF00452998

5. Bossavit, A., Computational Electromagnetism. Variational Formulation, Complementarity, Edge Elements, No. 2 in Academic Press Electromagnetism Series, Academic Press, San Diego, 1998.

6. Bossavit, A., "How weak is the weak solution in finite element methods?," IEEE Trans. Magnetics, Vol. MAG-34, 2429-2432, 1998.
doi:10.1109/20.717558

7. Bossavit, A., "On the geometry of electromagnetism IV: ‘Maxwell’s house’," J. Japan Soc. Appl. Electromagnetics & Mech., Vol. 6, 318-326, 1998.

8. Bossavit, A., "On the geometry of electromagnetism I: Affine space," J. Japan Soc. Appl. Electromagnetics & Mech., Vol. 6, 17-28, 1998.

9. Bossavit, A., "On the geometry of electromagnetism II: Geometrical objects," J. Japan Soc. Appl. Electromagnetics & Mech., Vol. 6, 114-123, 1998.

10. Bossavit, A., "On the geometry of electromagnetism III: Integration, Stokes’, Faraday’s law," J. Japan Soc. Appl. Electromagnetics & Mech., Vol. 6, 233-240, 1998.

11. Bossavit, A., "Computational electromagnetism and geometry. Building a finite-dimensional “Maxwell’s house” I: Network equations," J. Japan Soc. Appl. Electromagnetics & Mech., Vol. 7, 150-159, 1999.

12. Bossavit, A., "Computational electromagnetism and geometry II: Network constitutive laws," J. Japan Soc. Appl. Electromagnetics & Mech., Vol. 7, 294-301, 1999.

13. Bossavit, A., "Generalized finite differences in computational electromagnetics,", this volume.

14. Bossavit, A. and L. Kettunen, "Yee–like schemes on a tetrahedral mesh with diagonal lumping," Int. J. Numer. Modelling, Vol. 12, 129-142, 1999.
doi:10.1002/(SICI)1099-1204(199901/04)12:1/2<129::AID-JNM327>3.0.CO;2-G

15. Bossavit, A. and L. Kettunen, "Yee-like schemes on staggered cellular grids: A synthesis between FIT and FEM approaches,", contribution to COMPUMAG ’99., 1999.
doi:10.1002/(SICI)1099-1204(199901/04)12:1/2<129::AID-JNM327>3.0.CO;2-G

16. Brenner, S. and R. Scott, Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, Springer-Verlag, New York, 1994.
doi:10.1007/978-1-4757-4338-8

17. Brezzi, F. and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, 1991.
doi:10.1007/978-1-4612-3172-1

18. Burke, W., Applied Differential Geometry, Cambridge University Press, Cambridge, 1985.
doi:10.1017/CBO9781139171786

19. Cartan, H., Formes Differentielles, Hermann, Paris, 1967.

20. Chew, W., "Electromagnetic theory on a lattice," J. Appl. Phys., Vol. 75, 4843-4850, 1994.
doi:10.1063/1.355770

21. Ciarlet, P., "The finite element method for elliptic problems," Studies in Mathematics and its Applications, Vol. 4, North-Holland, Amsterdam, 1978.

22. Ciarlet, Jr., P. and J. Zou, "Fully discrete finite element approaches for time-dependent Maxwell equations," Numer. Math., Vol. 82, 193-219, 1999.
doi:10.1007/s002110050417

23. De La Bourdonnay, A. and S. Lala, "Duality between finite elements and finite volumes and Hodge operator," Numerical Methods in Engineering ’96, 557-561, Wiley & Sons, Paris, 1996.

24. Dezin, A., Multidimensional Analysis and Discrete Models, CRC Press, Boca Raton, FL, USA, 1995.

25. Girault, V. and P. Raviart, Finite Element Methods for Navier- Stokes Equations, Springer-Verlag, Berlin, 1986.
doi:10.1007/978-3-642-61623-5

26. Hackbusch, W., "On first and second order box schemes," Computing, Vol. 41, 277-296, 1989.
doi:10.1007/BF02241218

27. Hiptmair, R., "Canonical construction of finite elements," Math. Comp., Vol. 68, 1325-1346, 1999.
doi:10.1090/S0025-5718-99-01166-7

28. Hyman, J. and S. Steinberg, "The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials," J. Comp. Phys., Vol. 132, 130-148, 1997.
doi:10.1006/jcph.1996.5633

29. Hyman, J. and M. Shashkov, "Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids," Applied Numerical Mathematics, Vol. 25, 413-442, 1997.
doi:10.1016/S0168-9274(97)00097-4

30. Hyman, J. and M. Shashkov, "Natural discretizations for the divergence, gradient, and curl on logically rectangular grids," International Journal of Computers & Mathematics with Applications, Vol. 33, 81-104, 1997.
doi:10.1016/S0898-1221(97)00009-6

31. Hyman, J. and M. Shashkov, "Mimetic discretizations for Maxwell’s equations," J. Comp. Phys., Vol. 151, 881-909, 1999.
doi:10.1006/jcph.1999.6225

32. Hyman, J. and M. Shashkov, "The orthogonal decomposition theorems for mimetic finite difference methods," SIAM Journal on Numerical Analysis, Vol. 36, 788-818, 1999.
doi:10.1137/S0036142996314044

33. Iwaniec, T., "Nonlinear differential forms,", Lectures notes of the International Summer School in Jyvaskyla, 1998 80, Department of Mathematics, University of Jyvaskyla, Jyvaskyla, Finland, 1999.

34. Lala, S. and A. de la Bourdonnaye, "A finite-element method for Maxwell system preserving Gauss laws and energy," Tech. Rep. RR-3557, INRIA, Sophia Antipolis, France, November 1998. Submitted to Numer. Math..

35. Mattiussi, C., "An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology," J. Comp. Phys., Vol. 9, 295-319, 1997.

36. Monk, P., "An analysis of N´ed´elec’s method for the spatial discretization of Maxwell’s equations," J. Comp. Appl. Math., Vol. 47, 101-121, 1993.
doi:10.1016/0377-0427(93)90093-Q

37. Monk, P. and E. Suli, "A convergence analysis of Yee’s scheme on nonuniform grids," SIAM J. Numer. Anal., Vol. 31, 393-412, 1994.
doi:10.1137/0731021

38. Nedelec, J., "Mixed finite elements in R3," Numer. Math., Vol. 35, 315-341, 1980.
doi:10.1007/BF01396415

39. Nicolaides, R., "Direct discretization of planar div-curl problems," SIAM J. Numer. Anal., Vol. 29, 32-56, 1992.
doi:10.1137/0729003

40. Nicolaides, R. and D.-Q. Wang, "Convergence analysis of a covolume scheme for Maxwell’s equations in three dimensions," Math. Comp., Vol. 67, 947-963, 1998.
doi:10.1090/S0025-5718-98-00971-5

41. Nicolaides, R. and X. Wu, "Covolume solutions of threedimensional div-curl equations," SIAM J. Numer. Anal., Vol. 34, 2195-2203, 1997.
doi:10.1137/S0036142994277286

42. Sacco, R. and F. Saleri, "Exponentially fitted shape functions for advection-dominated flow problems in two dimensions," J. Comput. Appl. Math., Vol. 67, 161-165, 1996.
doi:10.1016/0377-0427(95)00149-2

43. Schuhmann, R. and T. Weiland, "A stable interpolation technique for FDTD on non-orthogonal grids," Int. J. Numer. Model., Vol. 11, 299-306, 1998.
doi:10.1002/(SICI)1099-1204(199811/12)11:6<299::AID-JNM314>3.0.CO;2-A

44. Shashkov, M., Conservative Finite-Difference Methods on General Grids, CRC Press, Boca Raton, 1996.

45. Shashkov, M. and S. Steinberg, "Solving diffusion equations with rough coefficients in rough grids," J. Comp. Phys., Vol. 129, 383-405, 1996.
doi:10.1006/jcph.1996.0257

46. Shashkov, M., B. Swartz, and B. Wendroff, "Local reconstruction of a vector field from its components on the faces of grid cells," Journal of Computational Physics, Vol. 139, 406-408, 1998.
doi:10.1006/jcph.1997.5877

47. Tarhasaari, T., L. Kettunen, and A. Bossavit, "Some realizations of a discrete Hodge: A reinterpretation of finite element techniques," IEEE Trans. Mag., Vol. 35, 1494-1497, 1999.
doi:10.1109/20.767250

48. Teixeira, F. and W. Chew, "Lattice electromagnetic theory from a topological viewpoint," J. Math. Phys., Vol. 40, 169-187, 1999.
doi:10.1063/1.532767

49. Tonti, E., "On the geometrical structure of electromagnetism," Graviation, Electromagnetism and Geometrical Structures, G. Ferrarese (ed.), 281–308, Pitagora, Bologna, Italy, 1996.

50. van Rienen, U., "Finite integration technique on triangular grids revisited," Int. J. Numer. Model., Vol. 12, 107-128, 1999.
doi:10.1002/(SICI)1099-1204(199901/04)12:1/2<107::AID-JNM322>3.0.CO;2-2

51. Weiland, T., "Die Diskretisierung der Maxwell-Gleichungen," Phys. Bl., Vol. 42, 191-201, 1986.
doi:10.1002/phbl.19860420710

52. Weiland, T., "Time domain electromagnetic field computation with finite difference methods," Int. J. Numer. Modelling, Vol. 9, 295-319, 1996.
doi:10.1002/(SICI)1099-1204(199607)9:4<295::AID-JNM240>3.0.CO;2-8

53. Yee, K., "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas and Propagation, Vol. 16, 302-307, 1966.