Recent Advances in Rational Approximation Methods for Spectral Fractional Diffusion Problems

This survey presents an overview of recent developments in the analysis and numerical treatment of spectral fractional diffusion equations. Particular attention is devoted to efficient strategies for solving spectral fractional diffusion problems, including approaches based on rational approximation that enable efficient numerical realization of fractional powers of elliptic operators. Building on these approximations, we discuss adaptive finite element discretization techniques for polygonal domains, where singularities and geometric irregularities require carefully designed mesh refinement strategies. The survey also highlights the role of fractional diffusion operators in the preconditioning of coupled and multiphysics problems, where they can significantly improve the robustness and convergence of iterative solvers. Furthermore, we review recent results on maximum principles and monotonicity preservation for spectral fractional diffusion–reaction equations, which are essential for ensuring physically meaningful numerical solutions. Finally, we discuss current efforts aimed at improving robustness and computational efficiency through reduced and multilevel iteration methods. These approaches provide scalable algorithms for large-scale problems while maintaining accuracy and stability. The survey concludes by outlining several open problems and promising directions for future research in the numerical analysis of fractional diffusion models.