Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-19T15:27:06.776Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  03 June 2013

Farah Abdallah ,
Serge Nicaise ,
Julie Valein  and
Ali Wehbe
Farah Abdallah
Affiliation:
Universitéde Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, 59313 Valenciennes Cedex 9, France; Farah.Abdallah@meletu.univ-valenciennes.fr
Serge Nicaise
Affiliation:
Universitéde Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, 59313 Valenciennes Cedex 9, France. Serge.Nicaise@univ-valenciennes.fr
Julie Valein
Affiliation:
Institut Elie Cartan Nancy (IECN), Nancy-Université & INRIA (Project-Team CORIDA), 54506 Vandoeuvre-lès-Nancy Cedex France; Julie.Valein@iecn.u-nancy.fr
Ali Wehbe
Affiliation:
Université Libanaise, Ecole Doctorale des Sciences et de Technologie, Hadath, Beyrouth, Liban; ali.wehbe@ul.edu.lb
Get access

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.

Keywords

Stabilitywave equationnumerical approximations
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 844 - 887
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alabau, F., Cannarsa, P. and Komornik, V., Indirect internal stabilization of weakly coupled evolution equations. J. Evol. Equ. 2 (2002) 127–150. Google Scholar
H. Amann, Linear and Quasilinear Parabolic Problems: abstract linear theory,Springer-Verlag. Birkhäuser 1 (1995).
Ammari, K. and Tucsnak, M., Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM: COCV 6 (2001) 361–386. Google Scholar
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).
Baiocchi, C., Komornik, V. and Loreti, P., Ingham-Beurling type theorems with weakened gap conditions. Acta Math. Hungar. 97 (2002) 55–95. Google Scholar
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.
Bátkai, A., Engel, K.-J., Prüss, J. and Schnaubelt, R., Polynomial stability of operator semigroups. Math. Nachr. 279 (2006) 1425–1440. Google Scholar
Batty, C.J.K. and Duyckaerts, T., Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ. 8 (2008) 765–780. Google Scholar
Borichev, A. and Tomilov, Y., Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347 (2010) 455–478. Google Scholar
Castro, C. and Micu, S., Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method. Numer. Math. 102 (2006) 413–462. Google Scholar
Castro, C., Micu, S. and Münch, A., 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. Google Scholar
P. G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978).
K. Engel and R. Nagel, One-parameter semigroups for linear evolution equations. Encyclopedia of Mathematics and its Applications. Springer-Verlag, New York (2000).
Ervedoza, S., Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes. Numer. Math. 113 (2009) 377–415. Google Scholar
Ervedoza, S. and Zuazua, E., Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. 91 (2009) 20–48. Google Scholar
Glowinski, R., Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation. J. Comput. Phys. 103 (1992) 189–221. Google Scholar
Glowinski, R., Kinton, W. and Wheeler, M.F., A mixed finite element formulation for the boundary controllability of the wave equation. Internat. J. Numer. Methods Engrg. 27 (1989) 623–635. Google Scholar
Glowinski, R., Li, C.H. and Lions, J.-L., 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. Google Scholar
R. Glowinski and J.-L. Lions, Exact and approximate controllability for distributed parameter systems, in Acta numerica, Cambridge Univ. Press, Cambridge (1995) 159–333.
E. Hewitt and K. Stromberg, Real and Abstract Analysis. Springer-Verlag, New York (1965).
Huang, F.L., Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ. 1 (1985) 43–56. Google Scholar
Infante, J. A. and Zuazua, E., Boundary observability for the space semi-discretizations of the one-dimensional wave equation. ESAIM: M2AN 33 (1999) 407–438. Google Scholar
Ito, K. and Kappel, F., The Trotter-Kato theorem and approximation of PDEs. Math. Comput. 67 (1998) 21–44. Google Scholar
Latushkin, Y. and Shvydkoy, R., 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. Google Scholar
León, L. and Zuazua, E., 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. Google Scholar
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).
Münch, A., A uniformly controllable and implicit scheme for the 1-D wave equation. ESAIM: M2AN 39 (2005) 377–418. Google Scholar
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.
Nicaise, S. and Valein, J., Stabilization of second order evolution equations with unbounded feedback with delay. ESAIM: COCV 16 (2010) 420–456. Google Scholar
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Math. Sciences. Springer-Verlag, New York 44 (1983).
Ramdani, K., Takahashi, T. and Tucsnak, M., Uniformly exponentially stable approximations for a class of second order evolution equations—application to LQR problems. ESAIM: COCV 13 (2007) 503–527. Google Scholar
P.-A. Raviart and J.-M. Thomas, Introduction l’analyse des équations aux dérivies partielles. Dunod, Paris (1998).
Shi, D.-H. and Feng, D.-X., Characteristic conditions of the generation of C 0 semigroups in a Hilbert space. J. Math. Anal. Appl. 247 (2000) 356–376. Google Scholar
Tcheugoué Tébou, L.R. and 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 (2003) 563–598. Google Scholar
Zuazua, E., 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. Google Scholar