The recent developments in computational methods for complex physical and biological systems have shown a significant shift towards high-order and adaptive techniques, driven by the need for more accurate and efficient solutions. The field is witnessing a surge in the application of discontinuous Galerkin methods (DGM) and finite element methods (FEM) with adaptive meshing, particularly in problems involving dynamic interfaces and non-linear constitutive relationships. These methods are being tailored to handle specific challenges such as crack propagation, ion transport, and quantum mechanical calculations, often integrating novel stabilization and splitting strategies to enhance computational efficiency. Notably, the use of image-processing techniques for adaptive domain decomposition in damage mechanics models represents a promising interdisciplinary approach that could be extended to other non-linear problems. The advancements in these areas are not only improving the accuracy of numerical simulations but also making them more accessible for practical applications in engineering and science.

Noteworthy Papers:

• A novel splitting strategy for solving generalized eigenvalue problems in Kohn-Sham density functional theory significantly accelerates simulation efficiency.
• An image-based adaptive domain decomposition method for continuum damage models demonstrates a novel use of image processing to inform computational mechanics.