On the Quadratic Finite Element Approximation of 1-d Waves: Propagation, Observation and Control∗

We study the propagation, observation and control properties of the quadratic P2classical finite element semi-discretization of the 1-d wave equation on a bounded interval. 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 secondorder finite element approximation introduces with respect to the P1-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 the Fourier truncation method or 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 L2 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.