The current developments in the research area are significantly advancing numerical methods and computational techniques across various fields, particularly in fluid dynamics and elasticity. There is a notable trend towards the development of more robust and scalable solvers, as well as methods that can handle complex geometries and non-smooth data more effectively. For instance, new finite element methods are being proposed that extend the applicability of traditional methods to broader classes of meshes, including those with zero-measure elements. These methods often involve innovative modifications that ensure convergence and accuracy, even in challenging scenarios. Additionally, there is a growing focus on integrating data assimilation techniques with computational models to enhance the accuracy of simulations, especially in fields like cardiovascular modeling where precise boundary conditions are crucial. The use of stochastic methods and advanced filtering techniques is proving to be effective in refining model predictions by incorporating real-time data. Furthermore, the development of fast direct solvers is making significant strides in reducing computational complexity and improving parallel efficiency, which is critical for large-scale problems in wave scattering and other areas. Overall, the field is moving towards more adaptive, scalable, and data-driven approaches that promise to significantly enhance the capabilities of numerical simulations in complex and real-world applications.