The recent developments in the research area have shown a significant shift towards innovative numerical methods and efficient solvers for complex problems in various fields. A notable trend is the emphasis on entropy-conservative and entropy-stable methods, which have been extended to new domains such as cut meshes and magnetohydrodynamic equations. These methods ensure stability and accuracy, even in the presence of complex boundary conditions and high anisotropy. Additionally, there is a growing interest in matrix-free implementations and non-nested multigrid methods, which offer flexibility and scalability for large-scale problems. The use of high-order finite element methods and discontinuous Galerkin approximations has also been advanced, with a focus on error estimates and conservation properties. Furthermore, novel preconditioning techniques and hierarchical solvers are being developed to handle pointwise inequality constraints and improve convergence rates in eigenvalue problems. Overall, the field is progressing towards more robust, efficient, and high-order accurate numerical solutions for a wide range of challenging problems.