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We study the propagation, observation and control properties of the 1-d waveequation on a bounded interval discretized in space using the quadraticclassical finite element approximation. A careful Fourier analysis of thediscrete wave dynamics reveals two different branches in the spectrum: theacoustic one, of physical nature, and the optic one, related to theperturbations that this second-order finite element approximation introduceswith respect to the linear one. On both modes there are high frequencies withvanishing group velocity as the mesh size tends to zero. This shows that theclassical property of continuous waves of being observable from the boundaryfails 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 mechanismsbased on a bi-grid algorithm for which one can recover the uniformity of theobservability constant in a finite time and, consequently, the possibility tocontrol with uniformly bounded $L^2$ - controls appropriate projections of thesolutions. This also allows showing that, by relaxing the control requirement,the controls are uniformly bounded and converge to the continuous ones as themesh size tends to zero.

On the quadratic finite element approximation of 1-d waves: propagation, observation and control