The goal of this article is to analyze the observability properties for a space semi-discrete approximation scheme derived from a mixed finite element method of the 1d wave equation on nonuniform meshes. More precisely, we prove that observability properties hold uniformly with respect to the mesh-size under some assumptions, which, roughly, measures the lack of uniformity of the meshes, thus extending the work [Castro and Micu, Numer. Math. 102 (2006) 413-462] to nonuniform meshes. Our results are based on a precise description of the spectrum of the discrete approximation schemes on nonuniform meshes, and the use of Ingham's inequality. We also mention applications to the boundary null controllability of the 1d wave equation, and to stabilization properties for the 1d wave equation. We finally present some applications for the corresponding fully discrete schemes, based on recent articles by the author.

Keywords: spectrum, observability, wave equation, semi-discrete systems, controllability, stabilization

@article{COCV_2010__16_2_298_0, author = {Ervedoza, Sylvain}, title = {Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {298--326}, publisher = {EDP-Sciences}, volume = {16}, number = {2}, year = {2010}, doi = {10.1051/cocv:2008071}, mrnumber = {2654195}, zbl = {1192.35109}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2008071/} }

TY - JOUR AU - Ervedoza, Sylvain TI - Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 298 EP - 326 VL - 16 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2008071/ DO - 10.1051/cocv:2008071 LA - en ID - COCV_2010__16_2_298_0 ER -

%0 Journal Article %A Ervedoza, Sylvain %T Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 298-326 %V 16 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2008071/ %R 10.1051/cocv:2008071 %G en %F COCV_2010__16_2_298_0

Ervedoza, Sylvain. Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 2, pp. 298-326. doi : 10.1051/cocv:2008071. http://archive.numdam.org/articles/10.1051/cocv:2008071/

[1] Exponentially stable approximations of weakly damped wave equations, in Estimation and control of distributed parameter systems (Vorau, 1990), Internat. Ser. Numer. Math. 100, Birkhäuser, Basel (1991) 1-33. | Zbl

, and ,[2] A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114 (1994) 185-200. | Zbl

,[3] Spectral properties of distance matrices. J. Phys. A 36 (2003) 3595-3616. | Zbl

, and ,[4] Numerical methods for Hamiltonian PDEs. J. Phys. A 39 (2006) 5287-5320. | Zbl

and ,[5] Boundary controllability of a linear semi-discrete 1-d wave equation derived from a mixed finite element method. Numer. Math. 102 (2006) 413-462. | Zbl

and ,[6] 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. | Zbl

, and ,[7] A priori estimates for mixed finite element methods for the wave equations. Comput. Methods Appl. Mech. Engrg. 82 (1990) 205-222. | Zbl

, and ,[8] The rate at which energy decays in a damped string. Comm. Partial Differ. Equ. 19 (1994) 213-243. | Zbl

and ,[9] The rate at which energy decays in a string damped at one end. Indiana Univ. Math. J. 44 (1995) 545-573. | Zbl

and ,[10] Perfectly matched layers in 1-d: Energy decay for continuous and semi-discrete waves. Numer. Math. 109 (2008) 597-634. | Zbl

and ,[11] Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. (to appear). | Zbl

and ,[12] On the observability of time-discrete conservative linear systems. J. Funct. Anal. 254 (2008) 3037-3078. | Zbl

, and ,[13] Linear PDEs and numerical methods that preserve a multisymplectic conservation law. SIAM J. Sci. Comput. 28 (2006) 260-277 (electronic). | Zbl

, and ,[14] Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation. J. Comput. Phys. 103 (1992) 189-221. | Zbl

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

, and ,[16] Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal. Math. 46 (1989) 245-258. | Zbl

,[17] Boundary observability for the space semi discretizations of the 1-d wave equation. Math. Model. Num. Ann. 33 (1999) 407-438. | Numdam | Zbl

and ,[18] Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41 (1936) 367-379. | Zbl

,[19] Uniform controllability of semidiscrete approximations of parabolic control systems. Systems Control Lett. 55 (2006) 597-609. | Zbl

and ,[20] Équations des ondes amorties, in Séminaire sur les Équations aux Dérivées Partielles, 1993-1994, École Polytechnique, France (1994). | Zbl

,[21] Contrôlabilité exacte, Perturbations et Stabilisation de Systèmes Distribués, Tome 1 : Contrôlabilité exacte, RMA 8. Masson (1988). | Zbl

,[22] The effect of group velocity in the numerical analysis of control problems for the wave equation, in Mathematical and numerical aspects of wave propagation - WAVES 2003, Springer, Berlin (2003) 195-200. | Zbl

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

,[24] Convergence of a multigrid method for the controllability of a 1-d wave equation. C. R. Math. Acad. Sci. Paris 338 (2004) 413-418. | Zbl

and ,[25] Wavelet filtering for exact controllability of the wave equation. SIAM J. Sci. Comput. 28 (2006) 1851-1885 (electronic). | Zbl

, and ,[26] Uniformly exponentially stable approximations for a class of second order evolution equations - application to LQR problems. ESAIM: COCV 13 (2007) 503-527. | Numdam | Zbl

, and ,[27] 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. | Zbl

and ,[28] Uniform boundary stabilization of the finite difference space discretization of the 1-d wave equation. Adv. Comput. Math. 26 (2007) 337-365. | Zbl

and ,[29] Group velocity in finite difference schemes. SIAM Rev. 24 (1982) 113-136. | Zbl

,[30] An introduction to nonharmonic Fourier series. Academic Press Inc., San Diego, CA, first edition (2001). | Zbl

,[31] Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square. J. Math. Pures Appl. 78 (1999) 523-563. | Zbl

,[32] Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197-243 (electronic). | Zbl

,*Cited by Sources: *