The current research landscape in the field is marked by significant advancements in numerical methods and computational techniques aimed at solving complex partial differential equations (PDEs) and inverse problems. There is a notable trend towards developing high-order and high-accuracy schemes, such as spectral element methods and finite difference schemes, which are being tailored for specific applications in quantum mechanics, electromagnetics, and scattering problems. These methods are increasingly incorporating innovative approaches like AI-enhanced approximations and novel boundary condition treatments to address the computational challenges posed by forward-peaked scattering and unbounded domains. Additionally, there is a growing interest in isogeometric analysis and collocation methods that leverage mixed-degree splines to enhance computational efficiency and accuracy in multi-patch domains. The field is also witnessing the application of higher-order methods in modeling complex anisotropic media, with a focus on both theoretical advancements and practical validation through numerical experiments. Overall, the research is progressing towards more sophisticated and efficient computational tools that promise to significantly advance the capabilities in modeling and solving intricate physical problems.

Noteworthy papers include one that introduces a new method for determining semilinear terms in elliptic equations from boundary measurements, demonstrating effective reconstruction using minimal data. Another notable contribution is the development of a higher-order spectral element method for electromagnetic modeling in complex anisotropic waveguides, which shows superior accuracy and efficiency compared to traditional methods.