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.

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}, pages = {844--887}, publisher = {EDP-Sciences}, volume = {19}, number = {3}, year = {2013}, doi = {10.1051/cocv/2012036}, mrnumber = {3092365}, zbl = {1275.65059}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2012036/} }

TY - JOUR AU - Abdallah, Farah AU - Nicaise, Serge AU - Valein, Julie AU - Wehbe, Ali TI - Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 844 EP - 887 VL - 19 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012036/ DO - 10.1051/cocv/2012036 LA - en ID - COCV_2013__19_3_844_0 ER -

%0 Journal Article %A Abdallah, Farah %A Nicaise, Serge %A Valein, Julie %A Wehbe, Ali %T Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 844-887 %V 19 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012036/ %R 10.1051/cocv/2012036 %G en %F 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/articles/10.1051/cocv/2012036/

[1] Indirect internal stabilization of weakly coupled evolution equations. J. Evol. Equ. 2 (2002) 127-150. | MR | Zbl

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

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

and ,[4] 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 | Zbl

and ,[5] Ingham-Beurling type theorems with weakened gap conditions. Acta Math. Hungar. 97 (2002) 55-95. | MR | Zbl

, and ,[6] 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 | Zbl

, and ,[7] Polynomial stability of operator semigroups. Math. Nachr. 279 (2006) 1425-1440. | MR | Zbl

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

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

and ,[10] 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 | Zbl

and ,[11] 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 | Zbl

, and ,[12] The finite element method for elliptic problems. North-Holland, Amsterdam (1978). | MR | Zbl

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

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

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

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

,[17] A mixed finite element formulation for the boundary controllability of the wave equation. Internat. J. Numer. Methods Engrg. 27 (1989) 623-635. | MR | Zbl

, and ,[18] 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

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

and ,[20] Real and Abstract Analysis. Springer-Verlag, New York (1965). | MR | Zbl

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

,[22] Boundary observability for the space semi-discretizations of the one-dimensional wave equation. ESAIM: M2AN 33 (1999) 407-438. | EuDML | Numdam | MR | Zbl

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

and ,[24] 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 | Zbl

and ,[25] 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. | EuDML | Numdam | MR | Zbl

and ,[26] 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 | Zbl

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

,[28] 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

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

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

,[31] Uniformly exponentially stable approximations for a class of second order evolution equations-application to LQR problems. ESAIM: COCV 13 (2007) 503-527. | EuDML | Numdam | MR | Zbl

, and ,[32] Introduction l'analyse des équations aux dérivies partielles. Dunod, Paris (1998).

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

and ,[34] 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 | Zbl

and ,[35] 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 | Zbl

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