The field of numerical methods for fluid dynamics and partial differential equations is experiencing significant developments, with a focus on improving the accuracy, efficiency, and robustness of various algorithms. One of the key directions is the development of new finite element methods, such as discontinuous Galerkin and mixed-degree local discontinuous Galerkin methods, which are being applied to a wide range of problems, including heterogeneous elliptic problems, Navier-Stokes equations, and multiphase Stokes problems. Another important area of research is the development of efficient multigrid solvers, which are being used to solve large-scale problems in fluid dynamics and other fields. Additionally, there is a growing interest in the development of adaptive methods, such as adaptive mesh-refinement and goal-oriented error estimation, which can help to improve the accuracy and efficiency of numerical simulations. Noteworthy papers in this area include the development of a structure preserving H-curl algebraic multigrid method for the eddy current equations, which demonstrates mesh independent convergence rates, and the design of efficient multigrid solvers for mixed-degree local discontinuous Galerkin multiphase Stokes problems, which show rapid convergence rates matching those of classical Poisson-style geometric multigrid methods.