Volume 55, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Page(s)||S969 - S991|
|Published online||26 February 2021|
Space time stabilized finite element methods for a unique continuation problem subject to the wave equation
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
2 Laboratoire de Mathématiques Blaise Pascal, Université Clermont Auvergne, UMR CNRS 6620, Campus des Cézeaux, Aubière 63177, France
* Corresponding author: firstname.lastname@example.org
Accepted: 25 August 2020
We consider a stabilized finite element method based on a spacetime formulation, where the equations are solved on a global (unstructured) spacetime mesh. A unique continuation problem for the wave equation is considered, where a noisy data is known in an interior subset of spacetime. For this problem, we consider a primal-dual discrete formulation of the continuum problem with the addition of stabilization terms that are designed with the goal of minimizing the numerical errors. We prove error estimates using the stability properties of the numerical scheme and a continuum observability estimate, based on the sharp geometric control condition by Bardos, Lebeau and Rauch. The order of convergence for our numerical scheme is optimal with respect to stability properties of the continuum problem and the approximation order of the finite element residual. Numerical examples are provided that illustrate the methodology.
Mathematics Subject Classification: 65M32 / 35R30
Key words: Unique continuation / data assimilation / wave equation / finite element method / geometric control / condition / observability estimate
© EDP Sciences, SMAI 2021
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