Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications

Farah Abdallah; Serge Nicaise; Julie Valein; Ali Wehbe

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 3, page 844-887
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Abdallah, Farah, et al. "Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications." ESAIM: Control, Optimisation and Calculus of Variations 19.3 (2013): 844-887. <http://eudml.org/doc/272848>.

@article{Abdallah2013,
abstract = {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.},
author = {Abdallah, Farah, Nicaise, Serge, Valein, Julie, Wehbe, Ali},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {stability; wave equation; numerical approximations; numerical viscosity term; second-order evolution equations; convergence},
language = {eng},
number = {3},
pages = {844-887},
publisher = {EDP-Sciences},
title = {Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications},
url = {http://eudml.org/doc/272848},
volume = {19},
year = {2013},
}

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
PB - EDP-Sciences
VL - 19
IS - 3
SP - 844
EP - 887
AB - 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.
LA - eng
KW - stability; wave equation; numerical approximations; numerical viscosity term; second-order evolution equations; convergence
UR - http://eudml.org/doc/272848
ER -

References

top
  1. [1] F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations. J. Evol. Equ.2 (2002) 127–150. Zbl1011.35018MR1914654
  2. [2] H. Amann, Linear and Quasilinear Parabolic Problems: abstract linear theory,Springer-Verlag. Birkhäuser 1 (1995). Zbl0819.35001MR1345385
  3. [3] K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM: COCV 6 (2001) 361–386. Zbl0992.93039MR1836048
  4. [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). Zbl0875.65087MR1115240
  5. [5] C. Baiocchi, V. Komornik and P. Loreti, Ingham-Beurling type theorems with weakened gap conditions. Acta Math. Hungar.97 (2002) 55–95. Zbl1012.42022MR1932795
  6. [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. Zbl0850.93719MR1155634
  7. [7] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups. Math. Nachr.279 (2006) 1425–1440. Zbl1118.47034MR2269247
  8. [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. Zbl1185.47043MR2460938
  9. [9] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups. Math. Ann.347 (2010) 455–478. Zbl1185.47044MR2606945
  10. [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. Zbl1102.93004MR2207268
  11. [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. Zbl1139.93005MR2387911
  12. [12] P. G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). Zbl0511.65078MR520174
  13. [13] K. Engel and R. Nagel, One-parameter semigroups for linear evolution equations. Encyclopedia of Mathematics and its Applications. Springer-Verlag, New York (2000). Zbl0952.47036MR1721989
  14. [14] S. Ervedoza, Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes. Numer. Math.113 (2009) 377–415. Zbl1170.93013MR2534130
  15. [15] S. Ervedoza and E. Zuazua, Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl.91 (2009) 20–48. Zbl1163.74019MR2487899
  16. [16] R. Glowinski, Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation. J. Comput. Phys.103 (1992) 189–221. Zbl0763.76042MR1196839
  17. [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. Zbl0711.65084MR1036928
  18. [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. Zbl0699.65055MR1039237
  19. [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. Zbl0838.93014MR1352473
  20. [20] E. Hewitt and K. Stromberg, Real and Abstract Analysis. Springer-Verlag, New York (1965). Zbl0137.03202MR367121
  21. [21] F.L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ.1 (1985) 43–56. Zbl0593.34048MR834231
  22. [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. Zbl0947.65101MR1700042
  23. [23] K. Ito and F. Kappel, The Trotter-Kato theorem and approximation of PDEs. Math. Comput.67 (1998) 21–44. Zbl0893.47025MR1443120
  24. [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. Zbl1036.47026MR1882702
  25. [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. Zbl1063.93025MR1932975
  26. [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). Zbl0924.73003MR1681343
  27. [27] A. Münch, A uniformly controllable and implicit scheme for the 1-D wave equation. ESAIM: M2AN 39 (2005) 377–418. Zbl1130.93016
  28. [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. Zbl1050.65068
  29. [29] S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay. ESAIM: COCV 16 (2010) 420–456. Zbl1217.93144MR2654201
  30. [30] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Math. Sciences. Springer-Verlag, New York 44 (1983). Zbl0516.47023MR710486
  31. [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. Zbl1126.93050MR2329173
  32. [32] P.-A. Raviart and J.-M. Thomas, Introduction l’analyse des équations aux dérivies partielles. Dunod, Paris (1998). 
  33. [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. Zbl1004.47026MR1769082
  34. [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. Zbl1033.65080MR2012934
  35. [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. Zbl0939.93016MR1697041

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.