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

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.