Convergence of a multi-grid algorithm for the controllability of the wave equation

In this paper we prove that the two-grid method proposed by Glowinski in [1] to numerically approximate the control of the wave equation converges in 1 — d. We prove this result in the context of the finite-element space semi-discretization. Our method of proof relies essentially on the particular properties of the Fourier representation of initial data of the coarse mesh when projected into the fine one. The explicit representation formula of the solutions shows that the high frequency components are modulated by some weights that diminish the effect of the spurious components. This fact, combined with discrete multipliers techniques, allows us to prove uniform observability inequalities. Classical arguments allow then proving the uniform boundedness of the controls and passing to the limit as the mesh size tends to zero. The controls obtained by this two-grid algorithm are not exact in the sense that they only guarantee the controllability of the projection of the solutions into the coarse mesh.