Boundary obeservability for the space semi-discretization for the 1-d wave equation

We consider space semi-discretizations of the 1- d wave equation in a bounded interval with homogeneous Dirichlet boundary conditions. We analyze the problem of boundary observability, i.e. , the problem of whether the total energy of solutions can be estimated uniformly in terms of the energy concentrated on the boundary as the net-spacing h → 0. We prove that, due to the spurious modes that the numerical scheme introduces at high frequencies, there is no such a uniform bound. We prove however a uniform bound in a subspace of solutions generated by the low frequencies of the discrete system. When h → 0 this finite-dimensional spaces increase and eventually cover the whole space. We thus recover the well-known observability property of the continuous system as the limit of discrete observability estimates as the mesh size tends to zero. We consider both finite-difference and finite-element semi-discretizations.