A finite element data assimilation method for the wave equation

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.

23 pages

References in corpus (2)

• Multiwave imaging in an enclosure with variable wave speed
• Inverse problems for linear hyperbolic equations using mixed formulations

Cited by in corpus (4)

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• Spacetime finite element methods for control problems subject to the wave equation
• A fully discrete numerical control method for the wave equation
• Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs