JournalsjemsVol. 11, No. 2pp. 351–391

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