Exterior problem of the Darwin model and its numerical computation
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 3, p. 515-532

In this paper, we study the exterior boundary value problems of the Darwin model to the Maxwell's equations. The variational formulation is established and the existence and uniqueness is proved. We use the infinite element method to solve the problem, only a small amount of computational work is needed. Numerical examples are given as well as a proof of convergence.

DOI : https://doi.org/10.1051/m2an:2003040
Classification:  35Q60,  65N30,  35J50
Keywords: Darwin model, Maxwell's equations, exterior problem, infinite element method
@article{M2AN_2003__37_3_515_0,
author = {Ying, Lung-An and Li, Fengyan},
title = {Exterior problem of the Darwin model and its numerical computation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {37},
number = {3},
year = {2003},
pages = {515-532},
doi = {10.1051/m2an:2003040},
zbl = {1031.35143},
mrnumber = {1994315},
language = {en},
url = {http://www.numdam.org/item/M2AN_2003__37_3_515_0}
}

Ying, Lung-An; Li, Fengyan. Exterior problem of the Darwin model and its numerical computation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 3, pp. 515-532. doi : 10.1051/m2an:2003040. http://www.numdam.org/item/M2AN_2003__37_3_515_0/

[1] P. Ciarlet Jr and J. Zou, Finite element convergence for the Darwin model to Maxwell's equations. Math. Modelling Numer. Anal. 31 (1997) 213-250. | Numdam | Zbl 0887.65121

[2] P. Degond and P.A. Raviart, An analysis of the Darwin model of approximation to Maxwell's equations. Forum Math. 4 (1992) 13-44. | Zbl 0755.35137

[3] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations. Springer, Berlin (1988). | MR 851383 | Zbl 0585.65077

[4] V. Girault and A. Sequeira, A well-posed problem for the exterior stokes equations in two and three dimensions. Arch. Ration. Mech. Anal. 114 (1991) 313-333. | Zbl 0731.35078

[5] D.W. Hewett and C. Nielson, A multidimensional quasineutral plasma simulation model. J. Comput. Phys. 29 (1978) 219-236. | Zbl 0388.76108

[6] O.A. Ladyzhenskaya,The Mathematical Theory of Viscous Incompressible Flow. 2nd ed., Gordon and Breach, New York (1969). | MR 254401 | Zbl 0184.52603

[7] T.-T. Li and T. Qin, Physics and Partial Differential Equations. Higher Education Press, Beijing (1997).

[8] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis. 3rd ed., North-Holland (1984). | MR 769654 | Zbl 0568.35002

[9] L.-A. Ying, Infinite element approximation to axial symmetric Stokes flow. J. Comput. Math. 4 (1986) 111-120. | Zbl 0598.76034

[10] L.-A. Ying, Infinite Element Methods. Peking University Press, Beijing and Vieweg and Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden (1995). | MR 1350539 | Zbl 0832.65120