Spacetime finite element methods for control problems subject to the wave equation

¹ Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
² Department of Mathematics, University of Toronto, 3359 Mississauga Road Deerfield Hall, Mississauga, Ontario, Canada L5L 1C6
³ Laboratoire de Mathématiques Blaise Pascal, Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France
⁴ Department of Mathematics and Statistics, University of Helsinki, P.O 68, 00014 University of Helsinki, Finland

^* Corresponding author: e.burman@ucl.ac.uk

Received: 18 October 2021
Accepted: 11 April 2023

Abstract

We consider the null controllability problem for the wave equation, and analyse a stabilized finite element method formulated on a global, unstructured spacetime mesh. We prove error estimates for the approximate control given by the computational method. The proofs are based on the regularity properties of the control given by the Hilbert Uniqueness Method, together with the stability properties of the numerical scheme. Numerical experiments illustrate the results.