Regular Article

Fully discrete finite element data assimilation method for the heat equation

¹ Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK.
² MRC Biostatistics Unit, University of Cambridge, Cambridge Biomedical Campus, Cambridge CB2 0SR, UK.
³ Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK.

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

Received: 15 August 2017
Accepted: 30 April 2018

Abstract

We consider a finite element discretization for the reconstruction of the final state of the heat equation, when the initial data is unknown, but additional data is given in a sub domain in the space time. For the discretization in space we consider standard continuous affine finite element approximation, and the time derivative is discretized using a backward differentiation. We regularize the discrete system by adding a penalty on the H²-semi-norm of the initial data, scaled with the mesh-parameter. The analysis of the method uses techniques developed in E. Burman and L. Oksanen [Numer. Math. 139 (2018) 505–528], combining discrete stability of the numerical method with sharp Carleman estimates for the physical problem, to derive optimal error estimates for the approximate solution. For the natural space time energy norm, away from t = 0, the convergence is the same as for the classical problem with known initial data, but contrary to the classical case, we do not obtain faster convergence for the L²-norm at the final time.