The field of numerical methods for partial differential equations is witnessing significant developments, with a focus on improving the accuracy and efficiency of solutions. Researchers are exploring new approaches to mesh generation, such as merged Voronoi-Delaunay meshes, to tackle complex geometries and anisotropic media. Additionally, there is a growing interest in fractional differential equations and their applications, with studies investigating the well-posedness and convergence properties of these equations. The use of data-driven methods, such as Diffusion Maps, is also becoming increasingly popular for approximating real-valued functions on smooth manifolds. Furthermore, novel coordinate transformations and high-order compact finite differences are being developed to solve PDEs on complex surfaces. Noteworthy papers in this area include:

• A paper on Learning functions through Diffusion Maps, which proposes a data-driven method for approximating real-valued functions on smooth manifolds, outperforming classical feedforward neural networks and interpolation methods.
• A paper on Unstructured to structured: geometric multigrid on complex geometries via domain remapping, which enables the use of robust geometric-style multigrid on complex domains via diffeomorphisms.