Fading regularization method for the stationary Stokes data assimilation problem

In this study, we address the ill-posed stationary Stokes data assimilation (DA) problem using the fading regularization method (FRM). It involves reconstructing the fluid velocity field throughout the study domain $\Omega$ in $R^n$, where $n$ is the dimension of the space, as well as the boundary conditions, using knowledge of some observations of the fluid velocity field $u^{obs}$ measured within a limited domain $\omega$ included in $\Omega$. Using the FRM, the main ill-posed problem is transformed into a sequence of well-posed constraint optimization problems and simplifies the resolution of DA problem. Additionally, we prove the convergence of both the continuous and the discrete formulations. This method is implemented numerically using the method of fundamental solutions (MFS) and several numerical simulations are shown to illustrate the performance of the algorithm in terms of efficiency, accuracy, convergence, stability, and robustness to noisy data, as well as its ability to deblur the data in $\omega$.