Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square

Sorin Micu ^{This paper studies the numerical approximation of the boundarycontrol for the wave equation in a square domain. It is knownthat the discrete and semi-discrete models obtained by discretizingthe wave equation with the usual finite-difference or finite-elementmethods do not provide convergent sequences of approximationsto the boundary control of the continuous wave equation as themesh size goes to zero. Here, we introduce and analyse a newsemi-discrete model based on the space discretization of thewave equation using a mixed finite-element method with two differentbasis functions for the position and velocity. The main theoreticalresult is a uniform observability inequality which allows usto construct a sequence of approximations converging to theminimal L2-norm control of the continuous wave equation. Wealso introduce a fully discrete system, obtained from our semi-discretescheme, for which we conjecture that it provides a convergentsequence of discrete approximations as both h and t, the timediscretization parameter, go to zero. We illustrate this factwith several numerical experiments.}