×

An analysis of Nédélec’s method for the spatial discretization of Maxwell’s equations. (English) Zbl 0784.65091

The method of J. C. Nédélec [Numer. Math. 35, 315-341 (1980; Zbl 0419.65069)] is slightly generalized. It is demonstrated that this method can be superconvergent at some special points. A convergence proof for the method of Kane S. Yee [Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propagation AP-14, 302-307 (1966)] is included.

MSC:

65Z05 Applications to the sciences
78M20 Finite difference methods applied to problems in optics and electromagnetic theory
65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
78A25 Electromagnetic theory (general)
35Q60 PDEs in connection with optics and electromagnetic theory

Citations:

Zbl 0419.65069
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bossavit, A., A rationale for “edge elements” in 3-D fields computations, IEEE Trans. Mag., 24, 74-79 (1988)
[2] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, 4 (1978), North-Holland: North-Holland Amsterdam, Stud. Math. Appl. · Zbl 0445.73043
[3] Ciarlet, P. G.; Raviart, A., A mixed finite element method for the biharmonic equation, (de Boor, C., Mathematical Aspects of Finite Elements in Partial Differential Equations (1974), Academic Press: Academic Press New York), 125-145 · Zbl 0337.65058
[4] Dubois, F., Discrete vector potential representation of a divergence free vector field in three dimensional domains: Numerical analysis of a model problem, SIAM J. Numer. Anal., 27, 1103-1142 (1990) · Zbl 0717.65086
[5] Duvaut, G.; Lions, J.-L., Inequalities in Mechanics and Physics (1976), Springer: Springer New York · Zbl 0331.35002
[6] Geveci, T., On the application of mixed finite element methods to the wave equation, Math. Model. Numer. Anal., 22, 243-250 (1988) · Zbl 0646.65083
[7] Girault, V., Incompressible finite element methods for Navier-Stokes equations with nonstandard boundary conditions in \(R^3\), Math. Comp., 51, 53-58 (1988) · Zbl 0666.76053
[8] Girault, V., Curl-conforming finite element methods for Navier-Stokes Equations with non-standard boundary conditions in \(R^3\), (Heywood, J. G., The Navier-Stokes Equations, Theory and Numerical Methods, 1431 (1990), Springer: Springer Berlin), 201-218, Lecture Notes in Math.
[9] Girault, V.; Raviart, P., Finite Element Methods for Navier-Stokes Equations (1986), Springer: Springer New York · Zbl 0585.65077
[10] Kikuchi, F., An isomorphic property of two Hilbert spaces appearing in electromagnetism: Analysis by the mixed formulation, Japan J. Appl. Math., 3, 53-58 (1986) · Zbl 0613.46040
[11] Kikuchi, F., Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism, Comput. Methods Appl. Mech. Engrg., 64, 509-521 (1987) · Zbl 0644.65087
[12] Lee, R. L.; Madsen, N. K., A mixed finite element formulation for Maxwell’s equations in the time domain, J. Comput. Phys., 88, 284-304 (1990) · Zbl 0703.65082
[13] Leis, R., Initial Boundary Value Problems in Mathematical Physics (1988), Wiley: Wiley New York · Zbl 0687.35065
[14] V. Levillain, Eigenvalue approximation by a mixed method for resonant inhomogeneous cavities with metallic boundaries, Preprint.; V. Levillain, Eigenvalue approximation by a mixed method for resonant inhomogeneous cavities with metallic boundaries, Preprint. · Zbl 0767.65091
[15] Monk, P., A mixed method for approximating Maxwell’s equations, SIAM J. Numer. Anal., 28, 1610-1634 (1991) · Zbl 0742.65091
[16] Monk, P., A comparison of three mixed methods for the time dependent Maxwell equations, SIAM J. Sci. Statist. Comput., 13, 1097-1122 (1992) · Zbl 0762.65081
[17] Monk, P., Analysis of a finite element method for Maxwell’s equations, SIAM J. Numer. Anal., 29, 714-729 (1992) · Zbl 0761.65097
[18] Monk, P., A finite element method for approximating the time-harmonic Maxwell equations, Numer. Math., 63, 243-261 (1992) · Zbl 0757.65126
[19] Monk, P., On the \(p\) and hp extension of Nédélec’s curl conforming elements, J. Comput. Appl. Math. (1994), to appear · Zbl 0820.65066
[20] Nédélec, J., Mixed finite elements in \(R^3\), Numer. Math., 35, 315-341 (1980) · Zbl 0419.65069
[21] Nédélec, J., Éléments finis mixtes incompressibles pour l’équation de Stokes dans \(R^3\), Numer. Math., 39, 97-112 (1982) · Zbl 0488.76038
[22] Neittaanmäki, P.; Saranen, J., Semi-discrete Galerkin approximation method applied to initial boundary value problems for Maxwell’s equations in anisotropic, inhomogeneous media, Proc. Roy. Soc. Edinburgh Sect. A, 89, 125-133 (1981) · Zbl 0469.65080
[23] Taflove, A.; Umashankar, K. R.; Beker, B.; Harfoush, F.; Yee, K. S., Detailed FD-TD analysis of electromagnetic fields penetrating narrow slots and lapped joints in thick conducting screens, IEEE Trans. Antennas and Propagation, AP-36, 247-257 (1988)
[24] Weiland, T., Numerical solution of Maxwell’s equation for static, resonant and transient problems, (Berceli, T., URSI International Symposium on Electromagnetic Theory, Part B, 28B (1986), North-Holland: North-Holland Amsterdam), 537-542, Stud. Electr. Electron. Engrg.
[25] Yee, K., Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas and Propagation, AP-16, 302-307 (1966) · Zbl 1155.78304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.