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Mathematics of Computation

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A finite element data assimilation method for the wave equation


Authors: Erik Burman, Ali Feizmohammadi and Lauri Oksanen
Journal: Math. Comp. 89 (2020), 1681-1709
MSC (2010): Primary 65M32, 65M60; Secondary 35R30, 65M12
DOI: https://doi.org/10.1090/mcom/3508
Published electronically: February 18, 2020
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Abstract: We design a primal-dual stabilized finite element method for the numerical approximation of a data assimilation problem subject to the acoustic wave equation. For the forward problem, piecewise affine, continuous, finite element functions are used for the approximation in space and backward differentiation is used in time. Stabilizing terms are added on the discrete level. The design of these terms is driven by numerical stability and the stability of the continuous problem, with the objective of minimizing the computational error. Error estimates are then derived that are optimal with respect to the approximation properties of the numerical scheme and the stability properties of the continuous problem. The effects of discretizing the (smooth) domain boundary and other perturbations in data are included in the analysis.


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Additional Information

Erik Burman
Affiliation: Department of Mathematics, University College London, London, UK-WC1E 6BT, United Kingdom
Email: e.burman@ucl.ac.uk

Ali Feizmohammadi
Affiliation: Department of Mathematics, University College London, London, UK-WC1E 6BT, United Kingdom
Email: a.feizmohammadi@ucl.ac.uk

Lauri Oksanen
Affiliation: Department of Mathematics, University College London, London, UK-WC1E 6BT, United Kingdom
Email: l.oksanen@ucl.ac.uk

DOI: https://doi.org/10.1090/mcom/3508
Received by editor(s): November 26, 2018
Received by editor(s) in revised form: September 16, 2019, and November 8, 2019
Published electronically: February 18, 2020
Additional Notes: The first author acknowledges funding by EPSRC grants EP/P01576X/1 and EP/P012434/1
The second author acknowledges funding by EPSRC grant EP/P01593X/1
The third author acknowledges funding by EPSRC grants EP/P01593X/1 and EP/R002207/1
Article copyright: © Copyright 2020 American Mathematical Society