Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications
ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 3, p. 844-887

In this paper, we consider the approximation of second order evolution equations. It is well known that the approximated system by finite element or finite difference is not uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Our goal is to damp the spurious high frequency modes by introducing numerical viscosity terms in the approximation scheme. With these viscosity terms, we show the exponential or polynomial decay of the discrete scheme when the continuous problem has such a decay and when the spectrum of the spatial operator associated with the undamped problem satisfies the generalized gap condition. By using the Trotter-Kato Theorem, we further show the convergence of the discrete solution to the continuous one. Some illustrative examples are also presented.

DOI : https://doi.org/10.1051/cocv/2012036
Classification:  65M60,  35L05,  35L15
Keywords: stability, wave equation, numerical approximations
@article{COCV_2013__19_3_844_0,
     author = {Abdallah, Farah and Nicaise, Serge and Valein, Julie and Wehbe, Ali},
     title = {Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {3},
     year = {2013},
     pages = {844-887},
     doi = {10.1051/cocv/2012036},
     zbl = {1275.65059},
     mrnumber = {3092365},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2013__19_3_844_0}
}
Abdallah, Farah; Nicaise, Serge; Valein, Julie; Wehbe, Ali. Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 3, pp. 844-887. doi : 10.1051/cocv/2012036. http://www.numdam.org/item/COCV_2013__19_3_844_0/

[1] F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations. J. Evol. Equ. 2 (2002) 127-150. | MR 1914654 | Zbl 1011.35018

[2] H. Amann, Linear and Quasilinear Parabolic Problems: abstract linear theory,Springer-Verlag. Birkhäuser 1 (1995). | MR 1345385 | Zbl 0819.35001

[3] K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM: COCV 6 (2001) 361-386. | Numdam | MR 1836048 | Zbl 0992.93039

[4] I. Babuska and J. Osborn, Eigenvalue problems, in Handbook of Numerical Analysis II Finite Element Methods. Edited by P.G. Ciarlet and J.L. Lions. North-Holland, Amsterdam (1991). | MR 1115240 | Zbl 0875.65087

[5] C. Baiocchi, V. Komornik and P. Loreti, Ingham-Beurling type theorems with weakened gap conditions. Acta Math. Hungar. 97 (2002) 55-95. | MR 1932795 | Zbl 1012.42022

[6] H. T. Banks, K. Ito and C. Wang, Exponentially stable approximations of weakly damped wave equations, in Estimation and control of distributed parameter systems (Vorau, 1990), Internat. Ser. Numer. Math., vol. 100. Birkhäuser, Basel (1991) 1-33. | MR 1155634 | Zbl 0850.93719

[7] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups. Math. Nachr. 279 (2006) 1425-1440. | MR 2269247 | Zbl 1118.47034

[8] C.J.K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ. 8 (2008) 765-780. | MR 2460938 | Zbl 1185.47043

[9] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347 (2010) 455-478. | MR 2606945 | Zbl 1185.47044

[10] C. Castro and S. Micu, Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method. Numer. Math. 102 (2006) 413-462. | MR 2207268 | Zbl 1102.93004

[11] C. Castro, S. Micu and A. Münch, Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square. IMA J. Numer. Anal. 28 (2008) 186-214. | MR 2387911 | Zbl 1139.93005

[12] P. G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). | MR 520174 | Zbl 0511.65078

[13] K. Engel and R. Nagel, One-parameter semigroups for linear evolution equations. Encyclopedia of Mathematics and its Applications. Springer-Verlag, New York (2000). | MR 1721989 | Zbl 0952.47036

[14] S. Ervedoza, Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes. Numer. Math. 113 (2009) 377-415. | MR 2534130 | Zbl 1170.93013

[15] S. Ervedoza and E. Zuazua, Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. 91 (2009) 20-48. | MR 2487899 | Zbl 1163.74019

[16] R. Glowinski, Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation. J. Comput. Phys. 103 (1992) 189-221. | MR 1196839 | Zbl 0763.76042

[17] R. Glowinski, W. Kinton and M.F. Wheeler, A mixed finite element formulation for the boundary controllability of the wave equation. Internat. J. Numer. Methods Engrg. 27 (1989) 623-635. | MR 1036928 | Zbl 0711.65084

[18] 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 1039237 | Zbl 0699.65055

[19] R. Glowinski and J.-L. Lions, Exact and approximate controllability for distributed parameter systems, in Acta numerica, Cambridge Univ. Press, Cambridge (1995) 159-333. | MR 1352473 | Zbl 0838.93014

[20] E. Hewitt and K. Stromberg, Real and Abstract Analysis. Springer-Verlag, New York (1965). | MR 367121 | Zbl 0137.03202

[21] F.L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ. 1 (1985) 43-56. | MR 834231 | Zbl 0593.34048

[22] J. A. Infante and E. Zuazua, Boundary observability for the space semi-discretizations of the one-dimensional wave equation. ESAIM: M2AN 33 (1999) 407-438. | Numdam | MR 1700042 | Zbl 0947.65101

[23] K. Ito and F. Kappel, The Trotter-Kato theorem and approximation of PDEs. Math. Comput. 67 (1998) 21-44. | MR 1443120 | Zbl 0893.47025

[24] Y. Latushkin and R. Shvydkoy, Hyperbolicity of semigroups and Fourier multipliers, in Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000), Oper. Theory Adv. Appl. 129 (2001) 341-363. | MR 1882702 | Zbl 1036.47026

[25] L. León and E. Zuazua, Boundary controllability of the finite-difference space semi-discretizations of the beam equation. ESAIM: COCV 8 (2002) 827-862. A tribute to J.L. Lions. | Numdam | MR 1932975 | Zbl 1063.93025

[26] Z. Liu and S. Zheng, Semigroups associated with dissipative systems, volume 398 of Chapman and Hall/CRC Research Notes in Mathematics. Chapman and Hall/CRC, Boca Raton, FL (1999). | MR 1681343 | Zbl 0924.73003

[27] A. Münch, A uniformly controllable and implicit scheme for the 1-D wave equation. ESAIM: M2AN 39 (2005) 377-418. | Numdam | Zbl 1130.93016

[28] M. Negreanu and E. Zuazua, A 2-grid algorithm for the 1-d wave equation, in Mathematical and numerical aspects of wave propagation-WAVES 2003. Springer, Berlin (2003) 213-217. | Zbl 1050.65068

[29] S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay. ESAIM: COCV 16 (2010) 420-456. | Numdam | MR 2654201 | Zbl 1217.93144

[30] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Math. Sciences. Springer-Verlag, New York 44 (1983). | MR 710486 | Zbl 0516.47023

[31] K. Ramdani, T. Takahashi and M. Tucsnak, Uniformly exponentially stable approximations for a class of second order evolution equations-application to LQR problems. ESAIM: COCV 13 (2007) 503-527. | Numdam | MR 2329173 | Zbl 1126.93050

[32] P.-A. Raviart and J.-M. Thomas, Introduction l'analyse des équations aux dérivies partielles. Dunod, Paris (1998).

[33] D.-H. Shi and D.-X. Feng, Characteristic conditions of the generation of C0 semigroups in a Hilbert space. J. Math. Anal. Appl. 247 (2000) 356-376. | MR 1769082 | Zbl 1004.47026

[34] 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. | MR 2012934 | Zbl 1033.65080

[35] E. Zuazua, Boundary observability for the finite-difference space semi-discretizations of the 2-d wave equation in the square. J. Math. pures et appl. 78 (1999) 523-563. | MR 1697041 | Zbl 0939.93016