Quasi exponential de ay of a nite dieren e

Niveau: Secondaire, Lycée, Terminale
Quasi exponential de ay of a nite dieren e spa e dis retization of the 1-d wave equation by pointwise interior stabilization Serge Ni aise, Julie Valein Université de Valen iennes et du Hainaut Cambrésis LAMAV, FR CNRS 2956 Institut des S ien es et Te hniques of Valen iennes F-59313 - Valen iennes Cedex 9 Fran e Julie.Valein,Serge.Ni aiseuniv-valen iennes.fr O tober 7, 2008 Abstra t We onsider the wave equation on an interval of length 1 with an interior damping at ?. It is well-known that this system is well-posed in the energy spa e and that its natural energy is dissipative. Moreover, as it was proved in [1?, the exponential de ay property of its solution is equivalent to an observability estimate for the orresponding onservative system. In this ase, the observability estimate holds if and only if ? is a rational number with an irredu tible fra tion ? = p q , where p is odd, and therefore under this ondition, this system is exponentially stable in the energy spa e. In this work, we are interested in the nite dieren e spa e semi- dis retization of the above system. As for other problems [24, 21?, we an expe t that the exponential de ay of this s heme does not hold in general due to high frequen y spurious modes.

• tion


• without any proof


• y??j ?


• tion ?


• interior damp- ing


• pointwise damping


• numeri al