Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization (Q2379350)

The authors consider the wave equation on an interval of length 1 with an interior damping at \(\xi \). For this case, the system is well-posed in the energy space and its natural energy is dissipative. Moreover, it was proved in \textit{K. Ammari, A. Henrot} and \textit{M. Tucsnak} [Asymptotic Anal. 28, No.~3--4, 215--240 (2001; Zbl 0999.35030)] that the exponential decay property of its solution is equivalent to an observability estimate for the corresponding conservative system, where the observability estimate holds if and only if \(\xi \) is a rational number with an irreducible fraction \(\xi =\frac{p}{q}\), where \(p\) is odd, and therefore under this condition, this system is exponentially stable in the energy space. In this work, they are interested in the finite difference space semi-discretization of the above system. For other cases \textit{E. Zuazua}, [SIAM Rev. 47, No.~2, 197--243 (2005; Zbl 1077.65095)]; \textit{L. T. Lebou} and \textit{E. Zuazua}, Adv. Comput. Math. 26, No.~1--3, 337--365 (2007; Zbl 1119.65086)], they show that the exponential decay of this scheme does not hold in general due to high frequency spurious modes, and then show that a filtering of high frequency modes allows to restore a quasi exponential decay of the discrete energy which is obtained based on a uniform interior observability estimate for filtered solutions of the corresponding conservative semi-discrete system.