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

This paper studies the numerical approximation of the boundary control for the wave equation in a square domain. It is known that the discrete and semi-discrete models obtained by discretizing the wave equation with the usual finite-difference or finite-element methods do not provide convergent sequences of approximations to the boundary control of the continuous wave equation as the mesh size goes to zero. Here, we introduce and analyse a new semi-discrete model based on the space discretization of the wave equation using a mixed finite-element method with two different basis functions for the position and velocity. The main theoretical result is a uniform observability inequality which allows us to construct a sequence of approximations converging to the minimal L²-norm control of the continuous wave equation. We also introduce a fully discrete system, obtained from our semi-discrete scheme, for which we conjecture that it provides a convergent sequence of discrete approximations as both h and Δt, the time discretization parameter, go to zero. We illustrate this fact with several numerical experiments.