Mimetic Discretizations for Maxwell's Equations

Journal of Computational Physics

Volumn 151, Issue 2, 1999, Pages 881-909

  Hyman, James M ^a   Shashkov, Mikhail ^a  

^a LOS ALAMOS NATIONAL LABORATORY   (United States)

Author keywords

Discrete vector analysis; Maxwell's equations; Mimetic finite difference methods; Numerical methods

Indexed keywords

COMPUTATION THEORY; DIFFERENTIATION (CALCULUS); FINITE DIFFERENCE METHOD; MATHEMATICAL OPERATORS; NUMERICAL METHODS;

DIFFERENTIAL OPERATORS; DISCRETE VECTOR; DISCRETE VECTOR ANALYSE; DISCRETIZATIONS; FINITE-DIFFERENCE METHODS; MIMETIC FINITE DIFFERENCE METHODS; MIMETICS; TENSOR CALCULUS; VECTOR ANALYSIS; VECTOR CALCULUS;

MAXWELL EQUATIONS;

EID: 0000397429     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcph.1999.6225     Document Type: Article

References (44)

1. Adam A. C., Serveniere A. G., Nedelec J. C., Raviart P. A. Study of an implicit scheme for integrating Maxwell's equations. Comput. Methods Appl. Mech. Eng. 22:1980;327.
2. Ardelyan N. V. The convergence of difference schemes for two-dimensional equations of acoustics and Maxwell's equations. USSR Comput. Math. Math. Phys. 23:1983;93.
3. Bossavit A. Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements. 1998.
4. Cai Z., Jones J. E., McCormick S. F., Russell T. F. Control-volume mixed finite element methods. Comput. Geosci. 1:1997;289.
5. Chew W. C. Electromagnetic theory on a lattice. J. Appl. Phys. 75:1994;4843.
6. Denisov A. A., Koldoba A. V., Poveshchenko Yu. A. The convergence to generalized solutions of difference schemes of the reference-operator method for Poisson's equation. USSR Comput. Math. Math. Phys. 29:1989;32.
7. M. V. Dmitrieva, A. A. Ivanov, V. F. Tishkin, A. P. Favorskii, Construction and investigation of support-operators finite-difference schemes for Maxwell equations in cylindrical geometry, Keldysh Inst. of Appl. Math. USSR Ac. of Sc. 27, 1985.
8. Girault V. Theory of a finite difference methods on irregular networks. SIAM J. Numer. Anal. 11:1974;260.
9. Gustafsson B., Kreiss H.-O., Oliger J. Time Dependent Problems and Difference Methods. 1995;445.
10. Hyman J. M., Shashkov M. Natural discretizations for the divergence, gradient, and curl on logically rectangular grids. Int. J. Comput. Math. Appl. 33:1997;81.
11. Hyman J. M., Shashkov M. The adjoint operators for the natural discretizations for the divergence, gradient, and curl on logically rectangular grids. IMACS J. Appl. Numer. Math. 25:1997;413.
12. Hyman J. M., Shashkov M. The orthogonal decomposition theorems for mimetic finite difference methods. SIAM J. Numer. Anal. 36:1998;788.
13. Hyman J. M., Shashkov M. The approximation of boundary conditions for mimetic finite difference methods. Int. J. Comput. Math. Appl. 36:1998;79.
14. J. M. Hyman, M. Shashkov, Mimetic Discretizations for Maxwell's Equations and Equations of Magnetic Diffusion, (http://cnls.lanl.gov/~shashkov) , Los Alamos, NM.
15. Hyman J. M., Shashkov M., Steinberg S. The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132:1997;130.
16. Jiang B., Wu J., Pavinelli L. A. The origin of spurious solutions in computational electromagnetics. J. Comput. Phys. 125:1996;104.
17. Jiang B. The Least-Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics. 1998;371.
18. Jin J. The Finite Element Method in Electromagnetics. 1993.
19. Korshiya T. K., Tishkin V. F., Favorskii A. P., Shashkov M. J. Flow-variational difference schemes for calculating the diffusion of a magnetic field. Sov. Phys. Dokl. 25:1980;832.
20. Lee R. L., Madsen N. K. A mixed finite element formulation for Maxwell's equations in the time domain. J. Comput. Phys. 88:1990;284.
21. Maikov A. R., Sveshnikov A. G., Yakunin S. A. Mathematical modeling of a microwave plasma generator. USSR Comput. Math. Math. Phys. 25:1985;149.
22. McCartin B. J., Dicello J. F. Three dimensional finite difference frequency domain scattering computation using the control region approximation. IEEE Trans. Magnetics. 25:1989;3092.
23. Monk P. A comparison of three mixed methods for the time-dependent Maxwell's equations. SIAM J. Sci. Stat. Comput. 13:1992;1097.
24. Monk P. Analysis of finite element method for Maxwell's equations. SIAM J. Numer. Anal. 29:1992;714.
25. Monk P. An analysis of Nedelec's method for the spatial discretization of Maxwell's equations. J. Comp. Appl. Math. 47:1993;101.
26. Morel J. E., Roberts R. M., Shashkov M. A local support-operators diffusion discretization scheme for quadrilateral r-z meshes. J. Comput. Phys. 144:1998;17.
27. 3. Numer. Math. 35:1980;315.
28. R. A. Nicolaides, A discrete vector field theory and some applications, Proceedings of IMACS'91 - 13th IMACS World Congress on Computation and Applied Mathematics, Trinity College, Dublin, Ireland, 1991, 120.
29. Nicolaides R. A., Wu X. Covolume solutions of three dimensional DIV-CURL equations. SIAM J. Numer. Anal. 34:1997;2195.
30. Nicolaides R. A., Wang D.-Q. Convergence analysis of covolume scheme for Maxwell's equations in three dimensions. Math. Comput. 67:1998;946.
31. Raviart P. A., Thomas J. M. A mixed finite element method for 2-nd order elliptic problems. Mathematical Aspects of Finite Element Methods. 1977;20.
32. Reitz J. R., Milford F. J., Christy R. W. Foundations of Electromagnetic Theory. 1980.
33. Samarskii A. A., Tishkin V. F., Favorskii A. P., Shashkov M. Yu. Operational finite-difference schemes. Differential Equations. 17:1981;854.
34. Schuhmann R., Weiland T. Stability of the FDTD algorithm on nonorthogonal grids related to the spatial interpolation scheme. IEEE Trans. Magnetics. 34:1998;2751.
35. Shashkov M. Conservative Finite-Difference Schemes on General Grids. 1995.
36. Shashkov M. Violation of conservation laws when solving difference equations by the iteration methods. U.S.S.R. Comput. Math. Math. Phys. 22:1982;131.
37. M. Shashkov, A. Solov'ov, Finite-difference schemes for solution of heat equation on Dirichlet grid, preprint, All Union Center for Mathematical Modeling, U.S.S.R. Academy of Sciences, 27, 1991.
38. Shashkov M., Steinberg S. Support-operator finite-difference algorithms for general elliptic problems. J. Comput. Phys. 118:1995;131.
39. Shashkov M., Steinberg S. Solving diffusion equations with rough coefficients in rough grids. J. Comput. Phys. 129:1996;383.
40. Finite Elements for Wave Electromagnetics: Methods and Techniques, P. P. SilvesterG. Pelosi, IEEE Press, New York, 1994.
41. Shercliff J. A. A Textbook of Magnetohydrodynamics. 1965;24.
42. Taflove A. Computational Electrodynamics: The Finite-Difference Time-Domain Method. 1995.
43. Weiland T. Time domain electromagnetic field computation with finite difference methods. Int. J. Numer. Model. 9:1996;295.
44. Yee K. S. Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE Trans. Antennas Propag. 14:1966;302.