Mesh adaptation for parameter calibration in fluid dynamics

We consider the calibration of parameters in physical models described by partial differential equations with applications in fluid dynamics. This task is formulated in a standard way as a constrained optimization problem with a cost functional of least squares type. The unknown parameters are calibrated using information obtained by measurements. An important issue in the numerical solution of this type of problem is the control of the errors introduced by discretization of the state equation. In this paper, we derive an a posteriori error estimator, which controls the discretization error with respect to a user-specified interest functional I. This interest functional (also called quantity of interest) might depend on both the state variable and the parameters and represents the goal of the computation. Our error estimator is derived in a general setting exploiting the special structure of the parameter calibration problem. It is used in an adaptive mesh refinement algorithm producing a sequence of economical meshes with respect to the quantity of interest. The proposed approach is illustrated on a numerical example involving the Navier-Stokes equations.