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.
Keywords: Unique continuation, data assimilation, wave equation, finite element method, geometric control, condition, observability estimate
@article{M2AN_2021__55_S1_S969_0,
author = {Burman, Erik and Feizmohammadi, Ali and M\"unch, Arnaud and Oksanen, Lauri},
title = {Space time stabilized finite element methods for a unique continuation problem subject to the wave equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {S969--S991},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {Suppl\'ement},
doi = {10.1051/m2an/2020062},
mrnumber = {4221321},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020062/}
}
TY - JOUR AU - Burman, Erik AU - Feizmohammadi, Ali AU - Münch, Arnaud AU - Oksanen, Lauri TI - Space time stabilized finite element methods for a unique continuation problem subject to the wave equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - S969 EP - S991 VL - 55 IS - Supplément PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020062/ DO - 10.1051/m2an/2020062 LA - en ID - M2AN_2021__55_S1_S969_0 ER -
%0 Journal Article %A Burman, Erik %A Feizmohammadi, Ali %A Münch, Arnaud %A Oksanen, Lauri %T Space time stabilized finite element methods for a unique continuation problem subject to the wave equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P S969-S991 %V 55 %N Supplément %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020062/ %R 10.1051/m2an/2020062 %G en %F M2AN_2021__55_S1_S969_0
Burman, Erik; Feizmohammadi, Ali; Münch, Arnaud; Oksanen, Lauri. Space time stabilized finite element methods for a unique continuation problem subject to the wave equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S969-S991. doi: 10.1051/m2an/2020062
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