In this paper we are concerned with the plane wave method for the discretization of time-harmonic Maxwell’s equations in three dimensions. As pointed out in Hiptmair et al. (Math. Comput. 82 (2013) 247–268), it is difficult to derive a satisfactory L2 error estimate of the standard plane wave approximation of the time-harmonic Maxwell’s equations. We propose a variant of the plane wave least squares (PWLS) method and show that the new plane wave approximations yield the desired L2 error estimate. Moreover, the numerical results indicate that the new approximations have sightly smaller L2 errors than the standard plane wave approximations. More importantly, the results are derived for more general models in inhomogeneous media.
Mots-clés : Maxwell’s equations, plane wave method, variational problem, L2 error estimate
@article{M2AN_2019__53_1_85_0,
author = {Hu, Qiya and Song, Rongrong},
title = {A variant of the plane wave least squares method for the time-harmonic {Maxwell{\textquoteright}s} equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {85--103},
publisher = {EDP-Sciences},
volume = {53},
number = {1},
year = {2019},
doi = {10.1051/m2an/2018043},
mrnumber = {3933916},
zbl = {1416.78026},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018043/}
}
TY - JOUR AU - Hu, Qiya AU - Song, Rongrong TI - A variant of the plane wave least squares method for the time-harmonic Maxwell’s equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2019 SP - 85 EP - 103 VL - 53 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018043/ DO - 10.1051/m2an/2018043 LA - en ID - M2AN_2019__53_1_85_0 ER -
%0 Journal Article %A Hu, Qiya %A Song, Rongrong %T A variant of the plane wave least squares method for the time-harmonic Maxwell’s equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2019 %P 85-103 %V 53 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018043/ %R 10.1051/m2an/2018043 %G en %F M2AN_2019__53_1_85_0
Hu, Qiya; Song, Rongrong. A variant of the plane wave least squares method for the time-harmonic Maxwell’s equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 1, pp. 85-103. doi : 10.1051/m2an/2018043. http://www.numdam.org/articles/10.1051/m2an/2018043/
[1] and , The Mathematical Theory of Finite Element Methods, 2nd edn. In Vol. 15 of Mathematics Applications Texts in Applied Mathematics. Springer-Verlag, New York (2002). | MR | Zbl
[2] and , Error estimates for the ultra weak variational formulation of the Helmholtz equation. ESAIM: M2AN 42 (2008) 925–940. | DOI | Numdam | MR | Zbl
[3] , Application d’une nouvelle formulation variationnelle aux équations d’ondes harmoniques, Problèmes de Helmholtz 2D et de Maxwell 3D. Ph.D. thesis, Université Paris IX Dauphine (1996).
[4] and , Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional helmholtz problem. SIAM J. Numer. Anal. 35 (1998) 255–299. | DOI | MR | Zbl
[5] , and , A survey of Trefftz methods for the Helmholtz equation. In: Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Springer International Publishing (2015) 237–279 | MR | Zbl
[6] , and , Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math. Comput. 82 (2013) 247–268. | DOI | MR | Zbl
[7] , and , Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J. Numer. Anal. 49 (2011) 264–284. | DOI | MR | Zbl
[8] , and , Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version. Found. Comput. Math. 16 (2016) 637–675. | DOI | MR | Zbl
[9] , and , Stability results for the time-harmonic Maxwell equations with impedance boundary conditions. Math. Mod. Meth. Appl. Sci. 21 (2011) 2263–2287. | DOI | MR | Zbl
[10] , and , Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes. Appl. Num. Math. 79 (2013) 79–91. | DOI | MR | Zbl
[11] and , A weighted variational formulation based on plane wave basis for discretization of Helmholtz equations. Int. J. Numer. Anal. Model. 11 (2014) 587–607. | MR | Zbl
[12] and , A plane wave least-squares method for time-harmonic Maxwell’s equations in absorbing media. SIAM J. Sci. Comput. 36 (2014) A1911–A1936. | MR | Zbl
[13] and , A plane wave method combined with local spectral elements for nonhomogeneous Helmholtz equation and time-harmonic Maxwell equations. Adv. Comput. Math. 44 (2018) 245–275. | DOI | MR | Zbl
[14] and , A Two-Steps Method based on Plane Wave for Nonhomogeneous Helmholtz Equations in Inhomogeneous Media. Numer. Math. Theor. Meth. Appl. 11 (2018) 453–476. | DOI | MR | Zbl
[15] , and , Solving Maxwell’s equations using the ultra weak variational formulation. J. Comput. Phys. 223 (2007) 731–758. | DOI | MR | Zbl
[16] , On Approximation in Meshless Methods, Frontiers of Numerical Analysis. Universitext. Springer, Berlin (2005) 65–141. | DOI | MR | Zbl
[17] and , A least-squares method for the helmholtz equation. Comput. Meth. Appl. Mech. Engng. 175 (1999) 121–136. | DOI | MR | Zbl
[18] , Tables of spherical codes (with collaboration of R.H. Hardin, W.D. Smith and others). Available at: http://www2.research.att.com/njas/packings (2000).
[19] , and , Parallel preconditioners for plane wave Helmholtz and Maxwell systems with large wave numbers, Int. J. Numer. Anal. Model. 13 (2016) 802–819. | MR | Zbl
Cité par Sources :
