Uniform stabilization of a viscous numerical approximation for a locally damped wave equation

Münch, Arnaud; Pazoto, Ademir Fernando

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 2, pp. 265-293.

This work is devoted to the analysis of a viscous finite-difference space semi-discretization of a locally damped wave equation in a regular 2-D domain. The damping term is supported in a suitable subset of the domain, so that the energy of solutions of the damped continuous wave equation decays exponentially to zero as time goes to infinity. Using discrete multiplier techniques, we prove that adding a suitable vanishing numerical viscosity term leads to a uniform (with respect to the mesh size) exponential decay of the energy for the solutions of the numerical scheme. The numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept, which guarantees that the low frequencies are damped correctly. Numerical experiments are presented and confirm these theoretical results. These results extend those by Tcheugoué-Tébou and Zuazua [Numer. Math. 95, 563-598 (2003)] where the 1-D case was addressed as well the square domain in 2-D. The methods and results in this paper extend to smooth domains in any space dimension.

[1] M. Asch and G. Lebeau, The spectrum of the damped wave operator for geometrically complex domain in ${ℝ}^2$. Experimental Math. 12 (2003) 227-241. | EuDML | MR | Zbl

[2] H.T. Banks, K. Ito and B. Wang, Exponentially stable approximations of weakly damped wave equations. Ser. Num. Math. 100 Birkhäuser (1990) 1-33. | MR | Zbl

[3] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization from the boundary. SIAM J. Control Opt. 30 (1992) 1024-1065. | MR | Zbl

[4] D. Chenais, On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975) 189-219. | MR | Zbl

[5] G.C. Cohen, Higher-order Numerical Methods for Transient Wave Equations. Scientific Computation, Springer (2002). | MR | Zbl

[6] C.M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity. Arch. Rational Mech. Anal. 29 (1968) 241-271. | MR | Zbl

[7] R. Glowinski, C.H. Li and J.-L. Lions, A numerical approach to the exact boundary controllability of the wave equation (I). Dirichlet Controls: Description of the numerical methods. Japan. J. Appl. Math. 7 (1990) 1-76. | MR | Zbl

[8] A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations. J. Differential Equations 59 (1985) 145-154. | MR | Zbl

[9] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portug. Math. 46 (1989) 245-258. | EuDML | MR | Zbl

[10] A. Henrot, Continuity with respect to the domain for the Laplacian: a survey. Control Cybernetics 23 (1994) 427-443. | Zbl

[11] J.A. Infante and E. Zuazua, Boundary observability for the space-discretizations of the 1-D wave equation. ESAIM: M2AN 33 (1999) 407-438. | Numdam | Zbl

[12] V. Komornik, Exact Controllability and Stabilization - The Multiplier Method. J. Wiley and Masson (1994). | MR | Zbl

[13] S. Krenk, Dispersion-corrected explicit integration of the wave equation. Comput. Methods Appl. Mech. Engrg. 191 (2001) 975-987. | Zbl

[14] J. Lagnese, Control of wave processes with distributed control supported on a subregion. SIAM J. Control Opt. 21 (1983) 68-85. | Zbl

[15] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod-Gauthier-Villars, Paris (1969). | MR | Zbl

[16] J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968). | MR | Zbl

[17] A. Münch, A uniformly controllable and implicit scheme for the 1-D wave equation. ESAIM: M2AN 39 (2005) 377-418. | Numdam | Zbl

[18] M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation. Israel J. Math. 95 (1996) 25-42. | Zbl

[19] M. Negreanu and E. Zuazua, Discrete Ingham inequalities and applications. C.R. Acad. Sci. Paris 338 (2004) 281-286. | Zbl

[20] O. Pironneau, Optimal shape design for elliptic systems. New York, Springer (1984). | MR | Zbl

[21] K. Ramdani, T. Takahashi and M. Tucsnak, Uniformly exponentially stable approximations for a class of second order evolution equations: Application to the optimal controle of flexible structures. Technical report, Prépublications de l'Institut Elie Cartan 27 (2003).

[22] M. Slemrod, Weak asymptotic decay via a “Relaxed Invariance Principle” for a wave equation with nonlinear, nonmonotone damping. Proc. Royal Soc. Edinburgh 113 (1989) 87-97. | Zbl

[23] L.R. Tcheugoué-Tébou, Stabilization of the wave equation with localized nonlinear damping. J. Differential Equations 145 (1998) 502-524. | Zbl

[24] L.R. Tcheugoué-Tébou and E. Zuazua, Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity. Numer. Math. 95 (2003) 563-598. | Zbl

[25] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping. Comm. Partial Differential Equation 15 (1990) 205-235. | Zbl

[26] E. Zuazua, Boundary observability for finite-difference space semi-discretizations of the 2-D wave equation in the square. J. Math. Pures Appl. 78 (1999) 523-563. | Zbl

[27] E. Zuazua, Optimal and approximate control of finite-difference approximation schemes for the 1-D wave equation. Rendiconti di Matematica, Serie VIII 24 (2004) 201-237. | Zbl

[28] E. Zuazua, Propagation, observation, control and numerical approximation of waves. SIAM Rev. 47 (2005) 197-243. | Zbl