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.

A Fully Discrete Numerical Control Method for the Wave Equation

“…The design of these terms is driven with the goal of minimizing the errors in the numerical approximation of the null controllability problem. The first two terms in \scrJ 1 correspond to the energy for the wave equation ( 1.3) at time t = T and seem a natural inclusion, while the remaining two terms are in part motivated by our previous works for data assimilation problems for heat and wave equations [6,7]. The regularization term in mixed derivatives also appears in [1].…”

“…This allows us to prove a discrete inf-sup property (Proposition 3.1), and subsequently an optimal convergence rate (Theorem 2.1). The present method can be seen as the continuation of our previous work in [6], where we studied numerical approximation of the dual problem to the controllability problem discussed here, that is, the data assimilation problem subject to the wave equation. A detailed comparison between these two works is given in section 6.…”

“…Next, we recall the following observability estimate originating from [3,4]. The formulation here is based on [16,Proposition 1.2] and is stated as it appears in [6,Theorem 2.2]. Theorem 1.2 (interior observability estimate).…”

“…In [6], we considered an approach based on finite element method for numerically solving the DA problem, using first order finite elements in space and finite differences in time. Analogous to the current work, this method was based on defining a discrete analogue for the Lagrangian \scrL that additionally incorporates numerical stabilizers in u. Optimal convergence rates were proven under the assumption that the geometric control condition is satisfied on the set \scrO (see [6,Theorem 4.6]).…”

“…Although our approach to solving the null controllability problem here draws similarities to that in [6], in the sense that similar numerical stabilization terms are used, we outline three of the key differences that makes the control problem more challenging.…”