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Stability analysis of the Interior Penalty Discontinuous Galerkin method for the wave equation

Published online by Cambridge University Press:  17 April 2013

Cyril Agut  and
Julien Diaz
Cyril Agut
Affiliation:
LMAP, University of Pau, INRIA Project-Team Magique-3D, France. cyril.agut@orange.fr
Julien Diaz
Affiliation:
INRIA Project-Team Magique-3D, LMAP, University of Pau, France
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Abstract

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.

Keywords

Discontinuous Galerkinpenalization coefficientCFL conditionwave equation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 3 , May 2013 , pp. 903 - 932
Copyright
© EDP Sciences, SMAI, 2013

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References

C. Agut and J. Diaz, Stability analysis of the interior penalty discontinuous Galerkin method for the wave equation. INRIA Res. Report (2010).
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).
Arnold, D.N., An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. Google Scholar
Arnold, D.N., Brezzi, F., Cockburn, B. and Marini, L.D., Unified analysis of disconitnuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749–1779. Google Scholar
C. Baldassari, Modélisation et simulation numérique pour la migration terrestre par équation d’ondes. Ph.D. Thesis (2009).
Benitez Alvarez, G., Dourado Loula, A.F., Dutrado Carmo, E.G. and Alves Rochinha, A., A discontinuous finite element formulation for Helmholtz equation. Comput. Methods. Appl. Mech. Engrg. 195 (2006) 4018–4035. Google Scholar
G. Cohen, Higher-Order Numerical Methods for Transient Wave Equations. Springer, Berlin (2001).
Cohen, S., Joly, P., Roberts, J.E. and Tordjman, N., Higher-order triangular finite elements with mass-lumping for the wave equation. SIAM J. Numer. Anal. 44 (2006) 2408–2431. Google Scholar
Cohen, S., Joly, P. and Tordjman, N., Higher-order finite elements with mass-lumping for the 1d wave equation. Finite Elem. Anal. Des. 16 (1994) 329–336. Google Scholar
Dablain, M.A., The application of high order differencing for the scalar wave equation. Geophys. 51 (1986) 54–56. Google Scholar
De Basabe, J.D. and Sen, M.K., 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. Google Scholar
Epshteyn, Y. and Rivière, B., Estimation of penalty parameters for symmetric interior penalty galerkin methods. J. Comput. Appl. Math. 206 (2007) 843–872. Google Scholar
S. Fauqueux, Eléments finis mixtes spectraux et couches absorbantes parfaitement adaptées pour la propagation d’ondes élastiques en régime transitoire. Ph.D. Thesis (2003).
Gilbert, J.-C. and Joly, P., Higher order time stepping for second order hyperbolic problems and optimal CFL conditions. Comput. Methods Appl. Sci. 16 (2008) 67–93. Google Scholar
Grote, M.J., Schneebeli, A. and Schötzau, D., Discontinuous Galerkin finite element method for the wave equation. SIAM J. Numer. Anal. 44 (2006) 2408–2431. Google Scholar
M.J. Grote and D. Schötzau, Convergence analysis of a fully discrete dicontinuous Galerkin method for the wave equation. Preprint No. 2008-04 (2008).
Komatitsch, D. and Tromp, J., Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys J. Int. 139 (1999) 806–822. Google Scholar
Lax, P. and Wendroff, B., Systems of conservation laws. Commun. Pure Appl. Math. XIII (1960) 217–237. Google Scholar
Seriani, G. and Priolo, E., Spectral element method for acoustic wave simulation in heterogeneous media. Finite Elem. Anal. Des. 16 (1994) 37–348. Google Scholar
Shahbazi, K., An explicit expression for the penalty parameter of the interior penalty method. J. Comput. Phys. 205 (2005) 401–407. Google Scholar
Shubin, G.R. and Bell, J.B., A modified equation approach to constructing fourth-order methods for acoustic wave propagation. SIAM J. Sci. Statist. Comput. 8 (1987) 135–151. Google Scholar
Warburton, T. and Hesthaven, J.S., On the constants in hp-finite element trace inverse inequalities. Comput. Methods Appl. Mech. Engrg. 192 (2003) 2765–2773. Google Scholar