The finite volume, finite element, and finite difference methods as numerical methods for physical field problems

Advances in Imaging and Electron Physics

Volumn 113, Issue C, 2000, Pages 1-146

  Mattiussi, Claudio ^a  

^a AREA Science Park   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 1542295883     PISSN: 10765670     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1076-5670(00)80012-9     Document Type: Article

References (93)

1. Abraham R., Marsden J.E., and Ratiu T. Manifolds, Tensor Analysis, and Applications (1988), Springer-Verlag, Berlin
2. Ahagon A., Fujiwara K., and Nakata T. Comparison of Various Kinds of Edge Elements for Electromagnetic Field Analysis. IEEE Trans. Magn. 32 (1996) 898-901
3. Albanese R., Fresa T., Martone R., and Rubinacci G. An Error Based Approach to the Solution of Full Maxwell Equations. IEEE Trans. Magn. 30 (1994) 2968-2971
4. Albanese R., and Rubinacci G. Analysis of Three-Dimensional Electromagnetic Fields Using Edge Elements. J. Comput. Phys. 108 (1993) 236-245
5. Albanese R., and Rubinacci G. Finite Element Methods for the Solution of 3D Eddy Current Problems. In: Hawkes P. (Ed). Advances in Imaging and Electron Physics 102 (1998) 1-86
6. Baldomir D., and Hammond P. Geometry of Electromagnetic Systems (1996), Oxford University Press, Oxford
7. Bamberg P., and Sternberg S. A Course in Mathematics for Students of Physics: 1-2 (1988), Cambridge University Press, Cambridge
8. Bateson G. The Logical Categories of Learning and Communication. Steps to an Ecology of Mind (1972), Ballantine Books, New York
9. Bellman R. Some Vistas of Modern Mathematics (1968), University of Kentucky Press
10. Belytschko T., Krongauz Y., Organ D., Fleming M., and Krysl P. Meshless Methods: An Overview and Recent Developments. Computer Methods in Applied Mechanics and Engineering 139 (1996) 3-47
11. Berenger J.P. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114 (1994) 185-200
12. Biro O., and Richter K.R. CAD in Electromagnetism. In: Hawkes P. (Ed). Advances in Electronics and Electron Physics 82 (1991) 1-96
13. Birss R.R. Multivector Analysis I-II. Physics Letters 78A (1980) 223-230
14. Bishop R.L., and Goldberg S.I. Tensor Analysis on Manifolds (1980), Dover, New York
15. Bodenmann S. Summation by Parts Formula for Noncentered Finite Differences, Research Report 95-07. Seminar für Angewandte Mathematik. ETH Zürich (1995)
16. Boothby W.M. An Introduction to Differentiable Manifolds and Riemannian Geometry. second edition (1986), Academic Press, San Diego
17. Bossavit A. Computational Electromagnetism (1998), Academic Press, San Diego
18. Bossavit A. How Weak Is the "Weak Solution" in Finite Element Methods?. IEEE Trans. Magn. 34 (1998) 2429-2432
19. Bowden K. On General Physical Systems Theories. Int. J. General Systems 18 (1990) 61-79
20. Branin Jr. F.H. The Algebraic-Topological Basis for Network Analogies and the Vector Calculus. Presented at the Symposium on Generalized Networks (1966), Polytechnic Institute of Brooklyn, New York
21. Burke W.L. Applied Differential Geometry (1985), Cambridge University Press, Cambridge
22. Burnett D.S. Finite Element Analysis (1987), Addison-Wesley, Reading, MA
23. Cartan E. Lecons sur les invariants intégraux (1922), Hermann, Paris
24. Cendes Z.J. Vector Finite Elements for Electromagnetic Field Computation. IEEE Trans. Magn. 27 (1991) 3958-3966
25. Choquet-Bruhat Y., and DeWitt-Morette C. Analysis, Manifolds and Physics (1977), North-Holland, Amsterdam
26. de Rham G. Sur l'Analysis situs des variétés à n dimensions. Journ. de Math. 10 (1931) 115-199
27. de Rham G. Variétés Différentiables (1960), Hermann, Paris
28. Deschamps G.A. Electromagnetics and Differential Forms. Proc. IEEE 69 no. 6 (1981) 676-696
29. Dezin A.A. Multidimensional Analysis and Discrete Models (1995), CRC Press, Boca Raton, FL
30. Dolcher M. Algebra Lineare (1978), Zanichelli, Bologna
31. Eilenberg S., and Steenrod N. Foundations of Algebraic Topology (1952), Princeton University Press, Princeton
32. Ferziger J.H., and Perić M. Computational Methods for Fluid Dynamics (1996), Springer-Verlag, Berlin
33. Flanders H. Differential Forms with Applications to the Physical Sciences (1989), Dover, New York
34. Fletcher C.A.J. Computational Galerkin Methods (1984), Springer-Verlag, Berlin
35. Franz W. Algebraic Topology (1968), Ungar, New York
36. Golias N.A., Tsiboukis T.D., and Bossavit A. Constitutive Inconsistency: Rigorous Solution of Maxwell Equations Based on a Dual Approach. IEEE Trans. Magn. 30 (1994) 3586-3589
37. Guibas L., and Stolfi J. Primitives for the Manipulation of General Subdivisions and the Computation of Voronoi, Diagrams. ACM Transactions on Graphics 4 (1985) 74-123
38. Gustafsson B., Kreiss H.-O., and Oliger J. Time Dependent Problems and Difference Methods (1995), Wiley, New York
39. Hocking J.G., and Young G.S. Topology (1988), Dover, New York
40. Hurewicz W., and Wallman H. Dimension Theory (1948), Princeton University Press, Princeton
41. Hyman J.M., and Shashkov M. Natural Discretization for the Divergence, Gradient, and Curl on Logically Rectangular Grids. Computers Math. Applic. 30 (1997) 81-104
42. Hyman J.M., and Shashkov M. Mimetic Discretization for Maxwell's Equations and Equations of Magnetic Diffusion. J. Comput. Phys. 151 (1999) 881-909
43. Isham C.J. Modern Differential Geometry for Physicists (1989), World Scientific, Singapore
44. Jackson J.D. Classical Electrodynamics (1975), Wiley, New York
45. Jin J. The Finite Element Method, in Electromagnetics (1993), Wiley, New York
46. Kunz K.S., and Luebbers R.J. Finite Difference Time Domain Method for Electromagnetics (1993), CRC Press, Boca Raton, FL
47. Lebesgue H. Leçons sur l'Intégration (1973), Chelsea, New York
48. Lee J., Lee R., and Cangellaris A. Time-Domain Finite Element Methods. IEEE Trans. Antennas Propagat 45 (1997) 430-442
49. Lele S.K. Compact Finite Difference Schemes with Spectral-like Resolution. J. Comput. Phys. 103 (1992) 16-42
50. Lilek Z., and Perić M. A Fourth-order Finite Volume Method with Colocated Variable Arrangement. Computers Fluids 24 (1995) 239-252
51. Mac Lane S. Mathematics Form and Function (1986), Springer-Verlag, Berlin
52. Madsen N.K. Divergence Preserving Discrete Surface Integral Methods for Maxwell's Curl Equations Using Non-orthogonal Unstructured Grids. J. Comput. Phys. 119 (1995) 34-45
53. Marmin F., Clénet S., Pirou F., and Bussy P. Error Estimation of Finite Element Solution in Non-Linear Magnetostatic 2D Problems. IEEE Trans. Magn. 34 (1998) 3268-3271
54. Mattiussi C. An Analysis of Finite Volume, Finite Element, and Finite Difference Methods Using Some Concepts from Algebraic Topology. J. Comput. Phys. 133 (1997) 289-309
55. Mattiussi C. Edge Elements and Cochain-Based Field Function Approximation. Proceedings of the 4th International Workshop on Electric and Magnetic Fields. Marseille (France) (1998) 301-306
56. Maxwell J.C. Remarks on the Mathematical Classification of Physical Quantities. Proc. of the London Math. Soc. 3 (1871) 224-232
57. Maxwell J.C. Traité Élémentaire d'Électricité (1884), Gauthier-Villars, Paris
58. Misner C.W., Thorne K.S., and Wheeler J.A. Gravitation (1970), Freeman, New York
59. Moore W. Schrödinger, Life and Thought (1989), Cambridge University Press, Cambridge
60. Mur G. Edge Elements, Their Advantages and Their Disadvantages. IEEE Trans. Magn. 30 (1994) 3552-3557
61. Nguyen D.B. Relativistic constitutive relations, differential forms, and the p-compound. Am. J. Phys. 60 (1992) 1134-1144
62. Oden J.T. Finite Element Applications in Mathematical Physics. In: Whiteman J.R. (Ed). The Mathematics of Finite Elements and Applications (1973), Academic Press, San Diego 239-282
63. Oden J.T., and Reddy J.N. Variational Methods in Theoretical Mechanics. second edition (1983), Springer-Verlag, Berlin
64. Oñate E., and Idelsohn S.R. A comparison between finite element and finite volume methods in CFD. Comput. Fluid Dyn. 1 (1992) 93
65. Palmer R.S., and Shapiro V. Chain models of physical behavior for engineering analysis and design. Computer Science Technical Report TR93-1375 (1993), Cornell University, New York
66. Penman J. Dual and Complementary Variational Techniques for the Calculation of Electromagnetic Fields. In: Hawkes P. (Ed). Advances in Electronics and Electron Physics 70 (1988) 315-364
67. Post E. Formal structure of Electromagnetics. General Covariance and Electromagnetics (1997), Dover, New York
68. Remacle J.-F., Geuzaine C., Dular P., Hedia H., and Legros W. Error Estimation Based on a New Principle of Projection and Reconstruction. IEEE Trans. Magn. 34 (1998) 3264-3267
69. Rikabi J., Bryant C.F., and Freeman E.M. An Error-based Approach to Complementary Formulations of Static Field Solutions. Int. J. Numer. Methods Eng. 26 (1988) 1963-1987
70. Schouten J.A. Tensor Analysis for Physicist (1989), Dover, New York
71. Schroeder W., and Wolff I. The Origin of Spurious Modes in Numerical Solutions of Electromagnetic Field Eigenvalue Problems. IEEE Trans. Microwave Theory Tech. 42 (1994) 644-653
72. Schutz B. Geometrical Methods of Mathematical Physics (1980), Cambridge University Press, Cambridge
73. Shashkov M. Conservative Finite-Difference Methods on General Grids (1996), CRC Press, Boca Raton FL
74. Shashkov M., and Steinberg S. Support-Operator Finite-Difference Algorithms for General Elliptic Problems. J. Comput. Phys. 118 (1995) 131-151
75. Strand B. Summation by Parts for Finite Difference Approximation for d/dx. J. Comput. Phys. 110 (1994) 47-67
76. Sun D., Manges J., Yuan X., and Cendes Z. Spurious Modes in Finite-Element Methods. IEEE Antennas and Propagation Magazine 37 (1995) 12-24
77. Taflove A. Computational Electrodynamics. The Finite-Difference Time-Domain Method (1995), Artech House, Boston
78. Taflove A. Advances in Computational Electrodynamics. The Finite-Difference Time-Domain Method (1998), Artech House, Boston, MA
79. Preprint, submitted to. Tarhasaari T., Kettunen L., and Bossavit A. An interpretation of the Galerkin method as the realization of a discrete Hodge operator. IEEE Trans. Magn. (1998)
80. Teixeira F.L., and Chew W.C. Differential Forms, Metrics and the Reflectionless Absorption of Electromagnetic Waves, Research Report CCEM-18-98 (1998), University of Illinois, Urbana
81. Teixeira F.L., and Chew W.C. Lattice electromagnetic theory from a topological viewpoint. J. Math. Phys. 40 (1999) 169-187
82. Tonti E. On the Formal Structure of Physical Theories (1975), Consiglio Nazionale delle Ricerche, Milano
83. Tonti E. The reason for analogies between physical theories. Appl. Math. Modelling 1 (1976) 37-50
84. Tonti E. Sulla struttura formale delle teorie fisiche. Rendiconti del Seminario Matematico e Fisico di Milano Vol. XLVI (1976) 163-257
85. Tonti E. Algebraic Topology and Computational Electromagnetism. Proceedings of the 4th International Workshop on Electric and Magnetic Fields. Marseille (France) (1998) 285-294
86. Truesdell C., and Toupin R.A. The classical field theories. In: Flugge S. (Ed). Handbuch der Physik Vol. 3/1 (1960), Springer-Verlag, Berlin 226-793
87. Truesdell C., and Noll W. The Non-Linear Field Theories of Mechanics. In: Flugge S. (Ed). Handbuch der Physik Vol. 3/3 (1965), Springer-Verlag, Berlin
88. Versteeg H.K., and Malalasekera W. An Introduction to Computational Fluid Dynamics. The Finite Volume Method (1995), Harlow, Longman, England
89. Warnick K.F., Selfridge R.H., and Arnold D.V. Teaching Electomagnetic Field Theory Using Differential Forms. IEEE Trans. on Education 40 (1997) 53-68
90. Webb J.P. Edge Elements and What They Can Do for You. IEEE Trans. Magn. 29 (1993) 1460-1465
91. Weiland T. On the Numerical Solution of Maxwell's Equations and Applications in the Field of Accelerator Physics. Particle Accelerators 15 (1984) 245-292
92. Weiland T. Time Domain Electromagnetic Field Computation with Finite Difference Methods. Int. J. Numer. Modelling 9 (1996) 295-319
93. Whitney H. Geometric Integration Theory (1957), Princeton University Press, Princeton