# Convergence of a two-grid algorithm for the control of the wave equation

### Enrique Zuazua

Universidad Autónoma de Madrid, Spain### Liviu I. Ignat

Universidad Autónoma de Madrid, Spain

## Abstract

We analyze the problem of boundary observability of the finite-difference space semi-discretizations of the 2-d wave equation in the square. We prove the uniform (with respect to the mesh-size) boundary observability for the solutions obtained by the two-grid preconditioner introduced by Glowinski \cite{0763.76042}. Our method uses previously known uniform observability inequalities for low frequency solutions and a dyadic spectral time decomposition. As a consequence we prove the convergence of the two-grid algorithm for computing the boundary controls for the wave equation. The method can be applied in any space dimension, for more general domains and other discretization schemes.

## Cite this article

Enrique Zuazua, Liviu I. Ignat, Convergence of a two-grid algorithm for the control of the wave equation. J. Eur. Math. Soc. 11 (2009), no. 2, pp. 351–391

DOI 10.4171/JEMS/153