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