Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization. (English) Zbl 1208.65131
The authors consider the wave equation on an interval of length 1 with an interior damping at \(\xi \). For this case, the system is well-posed in the energy space and its natural energy is dissipative. Moreover, it was proved in K. Ammari, A. Henrot and M. Tucsnak [Asymptotic Anal. 28, No. 3–4, 215–240 (2001; Zbl 0999.35030)] that the exponential decay property of its solution is equivalent to an observability estimate for the corresponding conservative system, where the observability estimate holds if and only if \(\xi \) is a rational number with an irreducible fraction \(\xi =\frac{p}{q}\), where \(p\) is odd, and therefore under this condition, this system is exponentially stable in the energy space. In this work, they are interested in the finite difference space semi-discretization of the above system. For other cases E. Zuazua, [SIAM Rev. 47, No. 2, 197–243 (2005; Zbl 1077.65095)]; L. T. Lebou and E. Zuazua, Adv. Comput. Math. 26, No. 1–3, 337–365 (2007; Zbl 1119.65086)], they show that the exponential decay of this scheme does not hold in general due to high frequency spurious modes, and then show that a filtering of high frequency modes allows to restore a quasi exponential decay of the discrete energy which is obtained based on a uniform interior observability estimate for filtered solutions of the corresponding conservative semi-discrete system.
Reviewer: Weizhong Dai (Ruston)
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35L05 | Wave equation |
References:
| [1] | Ammari, K., Henrot, A., Tucsnak, M.: Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string. Asymptot. Anal. 28(3–4), 215–240 (2001) · Zbl 0994.35030 |
| [2] | Banks, H.T., Ito, K., Wang, C.: Exponentially stable approximations of weakly damped wave equations. In: Estimation and Control of Distributed Parameter Systems (Vorau, 1990), Internat. Ser. Numer. Math., vol. 100, pp. 1–33. Birkhäuser, Basel (1991) · Zbl 0850.93719 |
| [3] | Brauer, U., Leugering, G.: On boundary observability estimates for semi-discretizations of a dynamic network of elastic strings. Control Cybern. 28(3), 421–447 (1999) (Recent advances in control of PDEs) · Zbl 0959.93003 |
| [4] | Castro, C., Micu, S.: Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method. Numer. Math. 102(3), 413–462 (2006) · Zbl 1102.93004 · doi:10.1007/s00211-005-0651-0 |
| [5] | Castro, C., Micu, S., Münch, A.: Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square. IMA J. Numer. Anal. 28(1), 186–214 (2008) · Zbl 1139.93005 · doi:10.1093/imanum/drm012 |
| [6] | Cazenave, T., Haraux, A.: Introduction Aux Problèmes d’Évolution Semi-linéaires, Mathématiques & Applications (Paris) [Mathematics and Applications], vol. 1. Ellipses, Paris (1990) · Zbl 0786.35070 |
| [7] | Glowinski, R.: Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation. J. Comput. Phys. 103(2), 189–221 (1992) · Zbl 0763.76042 · doi:10.1016/0021-9991(92)90396-G |
| [8] | Glowinski, R., Kinton, W., Wheeler, M.F.: A mixed finite element formulation for the boundary controllability of the wave equation. Int. J. Numer. Methods Eng. 27(3), 623–635 (1989) · Zbl 0711.65084 · doi:10.1002/nme.1620270313 |
| [9] | Glowinski, R., Li, C.H., Lions, J.-L.: A numerical approach to the exact boundary controllability of the wave equation. Dirichlet, I., controls: description of the numerical methods. Jpn. J. Appl. Math. 7(1), 1–76 (1990) · Zbl 0699.65055 · doi:10.1007/BF03167891 |
| [10] | Glowinski, R., Lions, J.-L.: Exact and approximate controllability for distributed parameter systems. In: Acta Numerica, 1995, Acta Numer., pp. 159–333. Cambridge University Press, Cambridge (1995) · Zbl 0838.93014 |
| [11] | Infante, J.A., Zuazua, E.: Boundary observability for the space semi-discretizations of the one-dimensional wave equation. M2AN 33, 407–438 (1999) · Zbl 0947.65101 · doi:10.1051/m2an:1999123 |
| [12] | Ingham, A.E.: Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41(1), 367–379 (1936) · Zbl 0014.21503 · doi:10.1007/BF01180426 |
| [13] | Isaacson, E., Keller, H.B.: Analysis of Numerical Methods. Wiley, New York (1966) · Zbl 0168.13101 |
| [14] | Komornik, V.: Exact Controllability and Stabilization. RAM: Research in Applied Mathematics. Masson, Paris (1994) (The multiplier method) · Zbl 0937.93003 |
| [15] | Lions, J.-L.: Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 8. Masson, Paris (1988) |
| [16] | Münch, A.: A uniformly controllable and implicit scheme for the 1-D wave equation. M2AN Math. Model. Numer. Anal. 39(2), 377–418 (2005) · Zbl 1130.93016 · doi:10.1051/m2an:2005012 |
| [17] | Negreanu, M., Zuazua, E.: Uniform boundary controllability of a discrete 1-D wave equation. Syst. Control. Lett. 48(3–4), 261–279 (2003) (Optimization and control of distributed systems) · Zbl 1157.93324 · doi:10.1016/S0167-6911(02)00271-2 |
| [18] | Nicaise, S., Valein, J.: Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks. Netw. Heterog. Media 2(3), 425–479 (2007) (electronic) · Zbl 1211.35050 · doi:10.3934/nhm.2007.2.425 |
| [19] | Rebarber, R.: Exponential stability of coupled beams with dissipative joints: a frequency domain approach. SIAM J. Control. Optim. 33(1):1–28 (1995) · Zbl 0819.93042 · doi:10.1137/S0363012992240321 |
| [20] | Tcheugoué Tébou, L.R., Zuazua, E.: Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity. Numer. Math. 95(3), 563–598 (2003) · Zbl 1033.65080 · doi:10.1007/s00211-002-0442-9 |
| [21] | Tcheugoué Tébou, L.R., Zuazua, E.: Uniform boundary stabilization of the finite difference space discretization of the 1-d wave equation. Adv. Comput. Math. 26, 337–365 (2007) · Zbl 1119.65086 · doi:10.1007/s10444-004-7629-9 |
| [22] | Zuazua, E.: Boundary observability for the finite-difference space semi-discretizations of the 2-d wave equation in the square. J. Math. Pures Appl. 78, 523–563 (1999) · Zbl 0939.93016 · doi:10.1016/S0021-7824(98)00008-7 |
| [23] | Zuazua, E.: Optimal and approximate control of finite-difference approximation schemes for the 1D wave equation. Rend. Mat. Appl. (7) 24(2), 201–237 (2004) · Zbl 1085.49041 |
| [24] | Zuazua, E.: Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47(2), 197–243 (2005) (electronic) · Zbl 1077.65095 · doi:10.1137/S0036144503432862 |
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