  We study the propagation, observation and control properties of the 1-d wave
equation on a bounded interval discretized in space using the quadratic
classical finite element approximation. A careful Fourier analysis of the
discrete wave dynamics reveals two different branches in the spectrum: the
acoustic one, of physical nature, and the optic one, related to the
perturbations that this second-order finite element approximation introduces
with respect to the linear one. On both modes there are high frequencies with
vanishing group velocity as the mesh size tends to zero. This shows that the
classical property of continuous waves of being observable from the boundary
fails to be uniform for this discretization scheme. As a consequence of this,
the controls of the discrete waves may blow-up as the mesh size tends to zero.
To remedy these high frequency pathologies, we design filtering mechanisms
based on a bi-grid algorithm for which one can recover the uniformity of the
observability constant in a finite time and, consequently, the possibility to
control with uniformly bounded $L^2$ - controls appropriate projections of the
solutions. This also allows showing that, by relaxing the control requirement,
the controls are uniformly bounded and converge to the continuous ones as the
mesh size tends to zero.
