Numerical simulation of wave propagation in inhomogeneous media using Generalized Plane Waves
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1387-1406.

The Trefftz Discontinuous Galerkin (TDG) method is a technique for approximating the Helmholtz equation (or other linear wave equations) using piecewise defined local solutions of the equation to approximate the global solution. When coefficients in the equation (for example, the refractive index) are piecewise constant it is common to use plane waves on each element. However when the coefficients are smooth functions of position, plane waves are no longer directly applicable. In this paper we show how Generalized Plane Waves (GPWs) can be used in a modified TDG scheme to approximate the solution for piecewise smooth coefficients in two dimensions. GPWs are approximate solutions to the equation that reduce to plane waves when the medium through which the wave propagates is constant. We shall show how to modify the TDG sesquilinear form to allow us to prove convergence of the GPW based version. The new scheme retains the high order convergence of the original TDG scheme (when the solution is smooth) and also retains the same number of degrees of freedom per element (corresponding to the directions of the GPWs). Unfortunately it looses the advantage that only skeleton integrals need to be performed. Besides proving convergence, we provide numerical examples to test our theory.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016067
Classification : 65N12, 65N15, 65N30
Mots clés : Trefftz based method, generalized plane waves, order of convergence
Imbert-Gérard, Lise-Marie 1 ; Monk, Peter 2

1 Courant Institute of Mathematical Sciences, New York University, 251 Mercer street, New York, NY 10012, USA.
2 Department of Mathematical Sciences, University of Delaware, Newark DE 19716, USA.
@article{M2AN_2017__51_4_1387_0,
     author = {Imbert-G\'erard, Lise-Marie and Monk, Peter},
     title = {Numerical simulation of wave propagation in inhomogeneous media using {Generalized} {Plane} {Waves}},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1387--1406},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {4},
     year = {2017},
     doi = {10.1051/m2an/2016067},
     mrnumber = {3702418},
     zbl = {1378.65179},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2016067/}
}
TY  - JOUR
AU  - Imbert-Gérard, Lise-Marie
AU  - Monk, Peter
TI  - Numerical simulation of wave propagation in inhomogeneous media using Generalized Plane Waves
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2017
SP  - 1387
EP  - 1406
VL  - 51
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2016067/
DO  - 10.1051/m2an/2016067
LA  - en
ID  - M2AN_2017__51_4_1387_0
ER  - 
%0 Journal Article
%A Imbert-Gérard, Lise-Marie
%A Monk, Peter
%T Numerical simulation of wave propagation in inhomogeneous media using Generalized Plane Waves
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2017
%P 1387-1406
%V 51
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2016067/
%R 10.1051/m2an/2016067
%G en
%F M2AN_2017__51_4_1387_0
Imbert-Gérard, Lise-Marie; Monk, Peter. Numerical simulation of wave propagation in inhomogeneous media using Generalized Plane Waves. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1387-1406. doi : 10.1051/m2an/2016067. http://www.numdam.org/articles/10.1051/m2an/2016067/

M. Bandres, Weber functions (parabolic cylinder functions). Available at http://www.mathworks.com/matlabcentral/fileexchange/46131-weber-functions–parabolic-cylinder-functions (2015).

M. Bandres, J. Gutiérrez-Vega and S. Chávez-Cerda, Parabolic nondiffracting optical fields. Opt. Lett. 29 (2004) 44–46. | DOI

A. Buffa and P. Monk, Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation. ESAIM: M2AN 42 (2008) 925–40. | DOI | Numdam | MR | Zbl

O. Cessenat, 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).

O. Cessenat and B. Després, Application of the ultra-weak variational formulation of elliptic PDEs to the 2-dimensional Helmholtz problem. SIAM J. Numer. Anal. 35 (1998) 255–99. | DOI | MR | Zbl

S. Esterhazy and J. Melenk, Numerical Analysis of Multiscale Problems, edited by I. Graham, T. Hou, O. Lakkis and R. Scheichl. Vol. 83 of Lect. Notes Comput. Sci. Eng. Springer (2012) 285–324. | MR | Zbl

C. Gittelson, R. Hiptmair and I. Perugia, Plane wave discontinuous Galerkin methods: analysis of the h-version. ESAIM: M2AN 43 (2009) 297–331. | DOI | Numdam | MR | Zbl

R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J. Numer. Anal. 49 (2011) 264–84. | DOI | MR | Zbl

R. Hiptmair, A. Moiola and I. Perugia, Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes. Appl. Numer. Math. 79 (2014) 79–91. | DOI | MR | Zbl

R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods: Exponential convergence of the hp-version, Found. Comput. Math. 16 (2015) 637–675. | DOI | MR | Zbl

T. Huttunen, P. Monk and J. Kaipio, Computational aspects of the Ultra Weak Variational Formulation. J. Comput. Phys. 182 (2002) 27–46. | DOI | MR | Zbl

L. Imbert-Gérard, Mathematical and numerical problems of some wave phenomena appearing in magnetic plasmas. Ph.D. thesis, Université Pierre et Marie Curie - Paris VI (2013).

L.-M. Imbert-Gérard, Interpolation properties of generalized plane waves. Numer. Math. 131 (2015) 683–711. | DOI | MR | Zbl

L.-M. Imbert-Gérard and B. Després, A generalized plane-wave numerical method for smooth nonconstant coefficients. IMA J. Numer. Anal. 34 (2014) 1072–1103. | DOI | MR | Zbl

S. Kapita, P. Monk and T. Warburton, Residual-based adaptivity and PWDG methods for the Helmholtz equation. SIAM J. Sci. Comput. 37 (2015) A1525–A1553. | DOI | MR | Zbl

T. Luostari, T. Huttunen and P. Monk, Improvements for the ultra weak variational formulation. Int. J. Numer. Meth. Eng. 94 (2013), 598–624. | DOI | Zbl

Cité par Sources :