New mapping properties of the time domain electric field integral equation
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 1-15.

We show some improved mapping properties of the Time Domain Electric Field Integral Equation and of its Galerkin semidiscretization in space. We relate the weak distributional framework with a stronger class of solutions using a group of strongly continuous operators. The stability and error estimates we derive are sharper than those in the literature.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016021
Classification : 65N30, 65N38, 65N12, 65N15, 78M15
Mots-clés : Electric field integral equation, retarded potentials, boundary integral equations, electromagnetic scattering, semigroup theory
Qiu, Tianyu 1 ; Sayas, Francisco-Javier 1

1 Department of Mathematical Sciences, University of Delaware, Newark DE 19716, USA.
@article{M2AN_2017__51_1_1_0,
     author = {Qiu, Tianyu and Sayas, Francisco-Javier},
     title = {New mapping properties of the time domain electric field integral equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1--15},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {1},
     year = {2017},
     doi = {10.1051/m2an/2016021},
     mrnumber = {3600998},
     zbl = {1360.78048},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2016021/}
}
TY  - JOUR
AU  - Qiu, Tianyu
AU  - Sayas, Francisco-Javier
TI  - New mapping properties of the time domain electric field integral equation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2017
SP  - 1
EP  - 15
VL  - 51
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2016021/
DO  - 10.1051/m2an/2016021
LA  - en
ID  - M2AN_2017__51_1_1_0
ER  - 
%0 Journal Article
%A Qiu, Tianyu
%A Sayas, Francisco-Javier
%T New mapping properties of the time domain electric field integral equation
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2017
%P 1-15
%V 51
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2016021/
%R 10.1051/m2an/2016021
%G en
%F M2AN_2017__51_1_1_0
Qiu, Tianyu; Sayas, Francisco-Javier. New mapping properties of the time domain electric field integral equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 1-15. doi : 10.1051/m2an/2016021. http://archive.numdam.org/articles/10.1051/m2an/2016021/

J. Ballani, L. Banjai, S. Sauter and A. Veit, Numerical solution of exterior Maxwell problems by Galerkin BEM and Runge-Kutta convolution quadrature. Numer. Math. 123 (2013) 643–670. | DOI | MR | Zbl

A. Bamberger and T.H. Duong, Formulation variationnelle espace-temps pour le calcul par potentiel retardé de la diffraction d’une onde acoustique. I. Math. Methods Appl. Sci. 8 (1986) 405–435. | DOI | MR | Zbl

A. Bamberger and T.H. Duong, Formulation variationnelle pour le calcul de la diffraction d’une onde acoustique par une surface rigide. Math. Methods Appl. Sci. 8 (1986) 598–608. | DOI | MR | Zbl

L. Banjai, A.R. Laliena and F.-J. Sayas, Fully discrete Kirchhoff formulas with CQ-BEM. IMA J. Numer. Anal. 35 (2015) 859–884. | DOI | MR | Zbl

A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell’s equations. I. An integration by parts formula in Lipschitz polyhedra. Math. Methods Appl. Sci. 24 (2001) 9–30. | DOI | MR | Zbl

A. Buffa and P. Ciarlet, Jr., On traces for functional spaces related to Maxwell’s equations. II. Hodge decompositions on the boundary of Lipschitz polyhedra and applications. Math. Methods Appl. Sci. 24 (2001) 31–48. | DOI | MR | Zbl

A. Buffa and R. Hiptmair, Galerkin boundary element methods for electromagnetic scattering. In Topics in computational wave propagation. Vol. 31 of Lect. Notes Comput. Sci. Eng. Springer, Berlin (2003) 83–124. | MR | Zbl

A. Buffa, M. Costabel and D. Sheen, On traces for 𝐇(𝐜𝐮𝐫𝐥,Ω) in Lipschitz domains. J. Math. Anal. Appl. 276 (2002) 845–867. | DOI | MR | Zbl

J.-C. Chan and P. Monk, Time dependent electromagnetic scattering by a penetrable obstacle. BIT Numer. Math. 55 (2015) 5–31. | DOI | MR | Zbl

Q. Chen, P. Monk, X. Wang and D. Weile, Analysis of convolution quadrature applied to the time-domain electric field integral equation. Commun. Comput. Phys. 11 (2012) 383. | DOI | MR | Zbl

M. Costabel, Time-dependent problems with the boundary integral equation method. In Vol. 1 of Encyclopedia of computational mechanics, edited by E. Stein, R. de Borst and T.J.R. Hughes. John Wiley & Sons, Ltd., Chichester (2004) 22. | MR

R. Dautray and J.-L. Lions, Evolution problems. I, With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon, Translated from the French by Alan Craig. Vol. 5 of Mathematical analysis and numerical methods for science and technology. Springer-Verlag, Berlin (1992). | MR | Zbl

V. Domínguez and F.-J. Sayas, Some properties of layer potentials and boundary integral operators for the wave equation. J. Integral Equations Appl. 25 (2013) 253–294. | DOI | MR | Zbl

R. Hiptmair and C. Schwab, Natural boundary element methods for the electric field integral equation on polyhedra. SIAM J. Numer. Anal. 40 (2002) 66–86. | DOI | MR | Zbl

S. Kesavan, Topics in functional analysis and applications. John Wiley & Sons Inc., New York (1989). | MR | Zbl

A.R. Laliena and F.-J. Sayas, A distributional version of Kirchhoff’s formula. J. Math. Anal. Appl. 359 (2009) 197–208. | DOI | MR | Zbl

J. Li, P. Monk and D. Weile, Time domain integral equation methods in computational electromagnetism. In Computational Electromagnetism, edited by A. Bermúdez de Castro and A. Valli. Vol. 2148 of Lect. Notes Math. Springer International Publishing (2015) 111–189. | MR

W. McLean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000). | MR | Zbl

M. Medvedik and Y. Smirnov, Ellipticity of the electric field integral equation for absorbing media and the convergence of the Rao-Wilton-Glisson method. Comput. Math. Math. Phys. 54 (2014) 114–122. | DOI | MR | Zbl

P. Monk, Finite Element Methods for Maxwell’s Equations. Numerical Mathematics and Scientific Computation. Clarendon Press (2003). | MR | Zbl

J.-C. Nédélec, Acoustic and electromagnetic equations, Integral representations for harmonic problems. Vol. 144 of Appl. Math. Sci. Springer-Verlag, New York (2001). | MR | Zbl

A. Pazy, Semigroups of linear operators and applications to partial differential equations. Vol. 44 of Appl. Math. Sci. Springer-Verlag, New York (1983). | MR | Zbl

T. Qiu and F. Sayas, The Costabel-Stephan system of boundary integral equations in the time domain. Math. Comput. (2015). | MR

S. Rao, D. Wilton and A. Glisson, Electromagnetic scattering by surfaces of arbitrary shape. Antennas Propag. IEEE Trans. 30 (1982) 409–418. | DOI

B.P. Rynne, The well-posedness of the electric field integral equation for transient scattering from a perfectly conducting body. Math. Methods Appl. Sci. 22 (1999) 619–631. | DOI | MR | Zbl

F.-J. Sayas, Energy estimates for Galerkin semidiscretizations of time domain boundary integral equations. Numer. Math. 124 (2013) 121–149. | DOI | MR | Zbl

F.-J. Sayas, Retarded potentials and time domain integral equations: a roadmap. Vol. 50 of Springer Series Comput. Math. Springer International Publishing (2016). | MR

I. Terrasse, Résolution mathématique et numérique des équations de Maxwell instationnaires par une méthode de potentiels retardés. Ph.D. thesis (1993).

F. Trèves, Topological vector spaces, distributions and kernels. Academic Press, New York-London (1967). | MR | Zbl

Cité par Sources :