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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.
Burman, Erik 1 ; Ish-Horowicz, Jonathan 1 ; Oksanen, Lauri 1
@article{M2AN_2018__52_5_2065_0,
author = {Burman, Erik and Ish-Horowicz, Jonathan and Oksanen, Lauri},
title = {Fully discrete finite element data assimilation method for the heat equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2065--2082},
publisher = {EDP-Sciences},
volume = {52},
number = {5},
year = {2018},
doi = {10.1051/m2an/2018030},
mrnumber = {3893359},
zbl = {1420.65124},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018030/}
}
TY - JOUR AU - Burman, Erik AU - Ish-Horowicz, Jonathan AU - Oksanen, Lauri TI - Fully discrete finite element data assimilation method for the heat equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 2065 EP - 2082 VL - 52 IS - 5 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018030/ DO - 10.1051/m2an/2018030 LA - en ID - M2AN_2018__52_5_2065_0 ER -
%0 Journal Article %A Burman, Erik %A Ish-Horowicz, Jonathan %A Oksanen, Lauri %T Fully discrete finite element data assimilation method for the heat equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 2065-2082 %V 52 %N 5 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018030/ %R 10.1051/m2an/2018030 %G en %F M2AN_2018__52_5_2065_0
Burman, Erik; Ish-Horowicz, Jonathan; Oksanen, Lauri. Fully discrete finite element data assimilation method for the heat equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 2065-2082. doi: 10.1051/m2an/2018030
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