Previous |  Up |  Next


acoustic wave equation; finite element method; Newmark method; new error estimate
We consider a family of conforming finite element schemes with piecewise polynomial space of degree $k$ in space for solving the wave equation, as a model for second order hyperbolic equations. The discretization in time is performed using the Newmark method. A new a priori estimate is proved. Thanks to this new a priori estimate, it is proved that the convergence order of the error is $h^{k}+\tau ^{2}$ in the discrete norms of $\mathcal {L}^{\infty }(0,T;\mathcal {H}^1(\Omega ))$ and $\mathcal {W}^{1,\infty }(0,T;\mathcal {L}^2(\Omega ))$, where $h$ and $\tau $ are the mesh size of the spatial and temporal discretization, respectively. These error estimates are useful since they allow us to get second order time accurate approximations for not only the exact solution of the wave equation but also for its first derivatives (both spatial and temporal). Even though the proof presented in this note is in some sense standard, the stated error estimates seem not to be present in the existing literature on the finite element methods which use the Newmark method for the wave equation (or general second order hyperbolic equations).
[1] Bernardi, C., Süli, E.: Time and space adaptivity for the second-order wave equation. Math. Models Methods Appl. Sci. 15 199-225 (2005). DOI 10.1142/S0218202505000339 | MR 2119997 | Zbl 1070.65083
[2] Brezis, H.: Analyse Fonctionnelle: Théorie et Applications. French Collection Mathématiques Appliquées pour la Maîtrise Masson, Paris (1983). MR 0697382 | Zbl 0511.46001
[3] H. F. Cooper, Jr.: Propagation of one-dimensional waves in inhomogeneous elastic media. SIAM Rev. 9 671-679 (1967). DOI 10.1137/1009108 | Zbl 0153.56403
[4] Dautray, R., Lions, J.-L.: Analyse Mathématique et Calcul Numérique Pour les Sciences et les Techniques. Vol. 9 French Masson, Paris (1988). MR 1016606 | Zbl 0652.45001
[5] Evans, L. C.: Partial Differential Equations. Graduate Studies in Mathematics 19 American Mathematical Society, Providence (1998). MR 1625845 | Zbl 0902.35002
[6] Feistauer, M., Felcman, J., Straškraba, I.: Mathematical and Computational Methods for Compressible Flow. Numerical Mathematics and Scientific Computation Oxford University Press, Oxford (2003). MR 2261900 | Zbl 1028.76001
[7] Grote, M. J., Schötzau, D.: Optimal error estimates for the fully discrete interior penalty DG method for the wave equation. J. Sci. Comput. 40 257-272 (2009). DOI 10.1007/s10915-008-9247-z | MR 2511734 | Zbl 1203.65182
[8] Karaa, S.: Finite element $\theta$-schemes for the acoustic wave equation. Adv. Appl. Math. Mech. 3 181-203 (2011). DOI 10.4208/aamm.10-m1018 | MR 2770084 | Zbl 1262.65131
[9] Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics 23 Springer, Berlin (2008). MR 1299729 | Zbl 1151.65339
[10] Raviart, P.-A., Thomas, J.-M.: Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles. French Collection Mathématiques Appliquées pour la Maîtrise Masson, Paris (1983). MR 0773854 | Zbl 0561.65069
[11] Zampieri, E., Pavarino, L. F.: Approximation of acoustic waves by explicit Newmark's schemes and spectral element methods. J. Comput. Appl. Math. 185 (2006), 308-325. DOI 10.1016/ | MR 2169068 | Zbl 1079.65093
Partner of
EuDML logo