Article
MSC:
35Q60,
65M70,
78A25,
78A40,
78M22 | MR 2476019 | Zbl 1212.78005 | DOI: 10.1007/s10492-009-0002-z
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
time-dependent electromagnetic field; cavity; vector and scalar potentials; Lorenz gauge; Chebyshev collocation
time-dependent electromagnetic field; cavity; vector and scalar potentials; Lorenz gauge; Chebyshev collocation
Summary:
The electromagnetic initial-boundary value problem for a cavity enclosed by perfectly conducting walls is considered. The cavity medium is defined by its permittivity and permeability which vary continuously in space. The electromagnetic field comes from a source in the cavity. The field is described by a magnetic vector potential ${\bf A}$ satisfying a wave equation with initial-boundary conditions. This description through ${\bf A}$ is rigorously shown to give a unique solution of the problem and is the starting point for numerical computations. A Chebyshev collocation solver has been implemented for a cubic cavity, and it has been compared to a standard finite element solver. The results obtained are consistent while the collocation solver performs substantially faster. Some time histories and spectra are computed.
References:
[1] Assous, F., P. Ciarlet, Jr., Labrunie, S., Segre, J.: Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: The singular complement method. J. Comput. Phys. 191 (2003), 147-176. DOI 10.1016/S0021-9991(03)00309-7 | MR 2008488 | Zbl 1033.65086
[2] Assous, F., Degond, P., Heintze, E., Raviart, P.-A., Segre, J.: On a finite-element method for solving the three-dimensional Maxwell equations. J. Comput. Phys. 109 (1993), 222-237. DOI 10.1006/jcph.1993.1214 | MR 1253460 | Zbl 0795.65087
[3] Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3: Spectral Theory and Applications. Springer Berlin (1990). MR 1064315
[4] Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5: Evolution Problems I. Springer Berlin (1992). MR 1156075 | Zbl 0755.35001
[5] Duvaut, G., Lions, J.-L.: Inequalities in Mechanics and Physics. Grundlehren der mathematischen Wissenschaften, 219. Springer Berlin-New York (1976). MR 0521262
[6] Evans, L. C.: Partial Differential Equations. Graduate Studies in Mathematics, 19. American Mathematical Society (AMS) Providence (1998). MR 1625845
[7] Hughes, T. J. R.: The Finite Element Method. Linear static and Dynamic Finite Element Analysis. Prentice-Hall Englewood Cliffs (1987). MR 1008473 | Zbl 0634.73056
[8] Jiang, B., Wu, J., Povinelli, L. A.: The origin of spurious solutions in computational electromagnetics. J. Comput. Phys. 125 (1996), 104-123. DOI 10.1006/jcph.1996.0082 | MR 1381806 | Zbl 0848.65086
[9] Ku, H. L., Hatziavramidis, D.: Chebyshev expansion methods for the solution of the extended Graetz problem. J. Comput. Phys. 56 (1984), 495-512. DOI 10.1016/0021-9991(84)90109-8 | MR 0768673 | Zbl 0572.76084
[10] Munz, C.-D., Omnes, P., Schneider, R., Sonnendrücker, E., Voss, U.: Divergence correction techniques for Maxwell solvers based on a hyperbolic model. J. Comput. Phys. 161 (2000), 484-511. DOI 10.1006/jcph.2000.6507 | MR 1764247
[11] Panofsky, W. K. H., Phillips, M.: Classical Electricity and Magnetism, 2nd ed. Addison-Wesley Reading (1962). MR 0135824 | Zbl 0122.21401
[12] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44. Springer New York (1983). MR 0710486
[13] Wloka, J.: Partial Differential Equations. Cambridge University Press Cambridge (1987). MR 0895589 | Zbl 0623.35006
[14] COMSOL$^{®}$, The COMSOL AB's homepage, www.comsol.com. Zbl 1220.65080
[15] MATLAB$^{®}$, The MathWorks, Inc. MATLAB's homepage, www.mathworks.com. Zbl 1234.92044
Search
Partner of
