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

Abstract

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 ob- tained by discretizing the wave equation with the usual ¯nite di®erence or ¯nite 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 (see [7, 15]). Here we introduce and analyze a new semi-discrete model based on the space discretization of the wave equa- tion using a mixed ¯nite element method with two di®erent 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 ¢t, the time discretization parameter, go to zero. We illustrate this fact with several numerical experiments.