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Finite element convergence for the Darwin model to Maxwell’s equations. (English) Zbl 0887.65121
The Darwin model of approximation to the Maxwell equations is considered. The solutions of elliptic boundary value problems in the Darwin model by finite element methods are analyzed. To this aim approximate variational formulations for the problem are derived, the well-posedness of the formulation is proved, finite element methods for the variational problems are proposed, the finite element convergence is shown, and error estimates are derived. The \(H(\text{curl}; \Omega)\) and \(H(\text{curl,div;} \Omega)\) variational formulations for the Darwin model approximations are derived.
Reviewer: V.Burjan (Praha)

MSC:
65Z05 Applications to the sciences
35Q60 PDEs in connection with optics and electromagnetic theory
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
78A25 Electromagnetic theory, general
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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References:
[1] R. A. ADAMS, 1975, Sobolev spaces. Academic Press, New York, 1975. Zbl0314.46030 MR450957 · Zbl 0314.46030
[2] J. J. AMBROSIANO, S. T. BRANDON and E. SONNENDRUCKER, 1995, A finite element formulation of the Darwin PIC model for use on unstructured grids J. Comput. Physics, 121(2), 281-297. Zbl0834.76052 MR1354305 · Zbl 0834.76052 · doi:10.1016/S0021-9991(95)90119-1
[3] I. BABUSKA, 1973, The finite element method with Lagrange multipliers Numer. Math, 20, 179-192. Zbl0258.65108 MR359352 · Zbl 0258.65108 · doi:10.1007/BF01436561 · eudml:132183
[4] M. BERCOVIER and O. PIRONNEAU, 1979, Error estimates for the finite element method solution of the Stokes problem in the primitive variables Numer. Math., 33, 211-224. Zbl0423.65058 MR549450 · Zbl 0423.65058 · doi:10.1007/BF01399555 · eudml:132638
[5] F. BREZZI, 1974, On the existence, uniqueness and approximation of saddle point problems arising from Lagrange multipliers. RAIRO Anal. Numer., 129-151. Zbl0338.90047 MR365287 · Zbl 0338.90047 · eudml:193255
[6] F. BREZZI and M. FORTIN, 1991, Mixed and hybrid finite element methods. Springer-Verlag, Berlin. Zbl0788.73002 MR1115205 · Zbl 0788.73002
[7] P. CIARLET, 1978, The finite element method for elliptic problems. North-Holland, Amsterdam. Zbl0383.65058 MR520174 · Zbl 0383.65058
[8] P. DEGOND and P. A. RAVIART, 1992, An analysis of the Darwin model of approximation to Maxwell’s equations Forum Math., 4, 13-44. Zbl0755.35137 MR1142472 · Zbl 0755.35137 · doi:10.1515/form.1992.4.13 · eudml:141662
[9] V. GIRAULT and P.-A. RAVIART, 1986, Finite element methods for Navier-Stokes equations. Springer-Verlag, Berlin. Zbl0585.65077 MR851383 · Zbl 0585.65077
[10] P. GRISVARD, 1985, Elliptic problems in nonsmooth domains. Pitman, Advanced Pubhshing Program, Boston. Zbl0695.35060 MR775683 · Zbl 0695.35060
[11] D. W. HEWETT and J. K. BOYD, 1987, Streamlined Darwin simulation of nonneutral plasmas. J. Comput. Phys., 73, 166-181. Zbl0611.76133 MR888935 · Zbl 0611.76133 · doi:10.1016/0021-9991(87)90007-6
[12] D. W. HEWETT and C. NIELSON, 1978, A multidimensional quasineutral plasma simulation model. J. Comput. Phys. 29, 219-236. Zbl0388.76108 · Zbl 0388.76108 · doi:10.1016/0021-9991(78)90153-5
[13] P. HOOD and G. TAYLOR, 1974, Navier-Stokes equation using mixed interpolation. In Oden, editor, Finite element methods in flow problems. UAH Press.
[14] J.-L. LIONS and E. MAGENES, 1968, Problèmes aux limites non homogènes et applications. Dunod, Paris. Zbl0165.10801 · Zbl 0165.10801
[15] J.-C. NEDELEC, 1980, Mixed finite éléments in R3. Numer. Math., 35, 315-341. Zbl0419.65069 MR592160 · Zbl 0419.65069 · doi:10.1007/BF01396415 · eudml:186293
[16] J.-C. NEDELEC, 1982, Eléments finis mixtes incompressibles pour l’équation de Stokes dans R3. Numer. Math., 39, 97-112. Zbl0488.76038 MR664539 · Zbl 0488.76038 · doi:10.1007/BF01399314 · eudml:132783
[17] C. NlELSON and H. R. LEWIS, 1976, Particle code models in the non radiative limit. Methods Comput. Phys., 16, 367-388.
[18] P.-A. RAVIART, 1993, Finite element approximation of the time-dependent Maxwell equations. Technical report, Ecole Polytechnique, France, GdR SPARCH #6.
[19] R. VERFURTH, 1984, Error estimates for a mixed finite element approximation of the Stokes equations. RAIRO Anal. Numer., 18(2), 175-182. Zbl0557.76037 MR743884 · Zbl 0557.76037 · eudml:193431
[20] C. WEBER, 1980, A local compactness theorem for Maxwell’s equations. Math. Meth. in the Appl. Sci., 2, 12-25. Zbl0432.35032 MR561375 · Zbl 0432.35032 · doi:10.1002/mma.1670020103
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