Stability analysis of the Interior Penalty Discontinuous Galerkin method for the wave equation
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 3, pp. 903-932.

We consider here the Interior Penalty Discontinuous Galerkin (IPDG) discretization of the wave equation. We show how to derive the optimal penalization parameter involved in this method in the case of regular meshes. Moreover, we provide necessary stability conditions of the global scheme when IPDG is coupled with the classical Leap-Frog scheme for the time discretization. Numerical experiments illustrate the fact that these conditions are also sufficient.

DOI : 10.1051/m2an/2012061
Classification : 35L05, 65M12, 65M60
Mots clés : discontinuous Galerkin, penalization coefficient, CFL condition, wave equation
@article{M2AN_2013__47_3_903_0,
     author = {Agut, Cyril and Diaz, Julien},
     title = {Stability analysis of the {Interior} {Penalty} {Discontinuous} {Galerkin} method for the wave equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {903--932},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {3},
     year = {2013},
     doi = {10.1051/m2an/2012061},
     mrnumber = {3056414},
     zbl = {1266.65151},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2012061/}
}
TY  - JOUR
AU  - Agut, Cyril
AU  - Diaz, Julien
TI  - Stability analysis of the Interior Penalty Discontinuous Galerkin method for the wave equation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2013
SP  - 903
EP  - 932
VL  - 47
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2012061/
DO  - 10.1051/m2an/2012061
LA  - en
ID  - M2AN_2013__47_3_903_0
ER  - 
%0 Journal Article
%A Agut, Cyril
%A Diaz, Julien
%T Stability analysis of the Interior Penalty Discontinuous Galerkin method for the wave equation
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2013
%P 903-932
%V 47
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2012061/
%R 10.1051/m2an/2012061
%G en
%F M2AN_2013__47_3_903_0
Agut, Cyril; Diaz, Julien. Stability analysis of the Interior Penalty Discontinuous Galerkin method for the wave equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 3, pp. 903-932. doi : 10.1051/m2an/2012061. http://www.numdam.org/articles/10.1051/m2an/2012061/

[1] C. Agut and J. Diaz, Stability analysis of the interior penalty discontinuous Galerkin method for the wave equation. INRIA Res. Report (2010).

[2] M. Ainsworth, P. Monk and W. Muniz, Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation. J. Sci. Comput. 27 (2006). | MR | Zbl

[3] D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742-760. | MR | Zbl

[4] D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of disconitnuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749-1779. | MR | Zbl

[5] C. Baldassari, Modélisation et simulation numérique pour la migration terrestre par équation d'ondes. Ph.D. Thesis (2009).

[6] G. Benitez Alvarez, A.F. Dourado Loula, E.G. Dutrado Carmo and A. Alves Rochinha, A discontinuous finite element formulation for Helmholtz equation. Comput. Methods. Appl. Mech. Engrg. 195 (2006) 4018-4035. | MR | Zbl

[7] G. Cohen, Higher-Order Numerical Methods for Transient Wave Equations. Springer, Berlin (2001). | MR | Zbl

[8] S. Cohen, P. Joly, J.E. Roberts and N. Tordjman, Higher-order triangular finite elements with mass-lumping for the wave equation. SIAM J. Numer. Anal. 44 (2006) 2408-2431. | Zbl

[9] S. Cohen, P. Joly and N. Tordjman, Higher-order finite elements with mass-lumping for the 1d wave equation. Finite Elem. Anal. Des. 16 (1994) 329-336. | MR | Zbl

[10] M.A. Dablain, The application of high order differencing for the scalar wave equation. Geophys. 51 (1986) 54-56.

[11] J.D. De Basabe and M.K. Sen, Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping. Geophys. J. Int. 181 (2010) 577-590.

[12] Y. Epshteyn and B. Rivière, Estimation of penalty parameters for symmetric interior penalty galerkin methods. J. Comput. Appl. Math. 206 (2007) 843-872. | MR | Zbl

[13] S. Fauqueux, Eléments finis mixtes spectraux et couches absorbantes parfaitement adaptées pour la propagation d'ondes élastiques en régime transitoire. Ph.D. Thesis (2003).

[14] J.-C. Gilbert and P. Joly, Higher order time stepping for second order hyperbolic problems and optimal CFL conditions. Comput. Methods Appl. Sci. 16 (2008) 67-93. | MR | Zbl

[15] M.J. Grote, A. Schneebeli and D. Schötzau, Discontinuous Galerkin finite element method for the wave equation. SIAM J. Numer. Anal. 44 (2006) 2408-2431. | MR | Zbl

[16] M.J. Grote and D. Schötzau, Convergence analysis of a fully discrete dicontinuous Galerkin method for the wave equation. Preprint No. 2008-04 (2008).

[17] D. Komatitsch and J. Tromp, Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys J. Int. 139 (1999) 806-822.

[18] P. Lax and B. Wendroff, Systems of conservation laws. Commun. Pure Appl. Math. XIII (1960) 217-237. | MR | Zbl

[19] G. Seriani and E. Priolo, Spectral element method for acoustic wave simulation in heterogeneous media. Finite Elem. Anal. Des. 16 (1994) 37-348. | MR | Zbl

[20] K. Shahbazi, An explicit expression for the penalty parameter of the interior penalty method. J. Comput. Phys. 205 (2005) 401-407. | Zbl

[21] G.R. Shubin and J.B. Bell, A modified equation approach to constructing fourth-order methods for acoustic wave propagation. SIAM J. Sci. Statist. Comput. 8 (1987) 135-151. | MR | Zbl

[22] T. Warburton and J.S. Hesthaven, On the constants in hp-finite element trace inverse inequalities. Comput. Methods Appl. Mech. Engrg. 192 (2003) 2765-2773. | MR | Zbl

Cité par Sources :