The field of finite element methods and discretization techniques is witnessing significant developments, with a focus on improving accuracy, efficiency, and applicability to complex problems. Researchers are exploring new finite element formulations, such as mixed-hybrid methods for Kirchhoff-Love shells, and developing innovative discretization techniques, including polytopal discretization methods and broken-FEEC frameworks for structure-preserving discretizations. The use of advanced numerical techniques, such as matrix-free methods and automatic code generation, is also gaining traction. Furthermore, the development of open-source libraries, such as PolyDiM, is facilitating the implementation and comparison of different methods. Noteworthy papers include:

• A novel mixed-hybrid finite element method for Kirchhoff-Love shells, which enables the use of classical, possibly higher-order Lagrange elements.
• The introduction of PolyDiM, a C++ library for polytopal discretization methods, which provides robust and modular tools for advanced numerical techniques.
• A broken-FEEC framework for structure-preserving discretizations of polar domains with tensor-product splines, which enables the use of standard tensor-product spline spaces while ensuring stability and smoothness for the solutions.