The field of numerical analysis is witnessing significant developments in the solution of complex fluid dynamics and inverse problems. Researchers are exploring new numerical methods and techniques to improve the accuracy and efficiency of simulations, particularly in the context of non-Newtonian fluids, thermally driven active fluids, and time-fractional diffusion problems. The use of mixed formulations, stabilized approaches, and iso-parametric finite element discretizations is becoming increasingly popular, as these methods offer improved stability and convergence properties. Additionally, the development of new algorithms and functional frameworks, such as the Kohn-Vogelius type functional, is enabling the reconstruction of potential parameters and the solution of inverse problems with greater accuracy and robustness. Noteworthy papers in this regard include the development of a convergent finite element method for two-phase Stokes flow driven by surface tension, which provides a fundamental numerical analysis tool for general curvature-driven free boundary problems. Another notable work is the reconstruction of a potential parameter in time-fractional diffusion problems via a Kohn-Vogelius type functional, which demonstrates the effectiveness and robustness of the proposed method through several numerical examples.