Fluid flow often interacts with complex media, evolving interfaces, and uncertain data—posing significant challenges for both modeling and computation. In this talk, I present recent advances in finite element methods that address these challenges through three complementary perspectives: coupling, splitting, and uncertainty quantification.

Coupling: A unified Banach space mixed formulation for fluid–poroelastic interaction, capturing the interplay between free flow and deformable porous media within a rigorous functional-analytic framework.

Splitting: A stable and efficient Robin–Robin partitioned method that overcomes classical instabilities in loosely-coupled schemes, enabling robust time-dependent simulations.

Uncertainty: A Nitsche-based approach for incorporating randomness in boundary conditions, together with reliable a posteriori error control.

Across these developments, the emphasis is on designing methods that balance mathematical rigor, physical fidelity, and computational efficiency. The results highlight how modern finite element techniques can provide stable, accurate, and flexible tools for complex fluid problems arising in science and engineering.