The field of numerical methods for partial differential equations is witnessing significant developments, with a focus on improving the accuracy and efficiency of solvers for complex problems. Researchers are exploring new approaches to tackle challenges such as boundary and interface conditions, high-order discretizations, and adaptive time stepping. One notable trend is the integration of multigrid methods with high-order immersed finite difference solvers, enabling the solution of large-scale problems on complex domains. Additionally, innovative regularization techniques are being proposed to solve ill-posed problems, such as Fredholm integral equations of the first kind. Noteworthy papers include: A high order multigrid-preconditioned immersed interface solver for the Poisson equation, which demonstrates the ability to solve high-order discretizations of Laplace and Poisson problems on complex 3D domains. A new iterated Tikhonov regularization method for Fredholm integral equation of first kind, which achieves optimal order under a-priori assumption and shows promising results in numerical experiments.