Current Trends in Multiscale and Fractional Differential Equations

The recent developments in the field of multiscale and fractional differential equations have shown a significant shift towards methods that do not rely on scale separation assumptions. Researchers are increasingly focusing on techniques that can handle rough coefficients and non-uniform temporal grids, which are common in real-world applications. The integration of implicit-explicit methods with mixed finite element techniques has proven effective for time-fractional partial integro-differential equations, offering stability and optimal error estimates. Additionally, there is a growing interest in error control methods for numerical interpolation in computational technologies, particularly in ensuring continuity and minimizing the Runge phenomenon.

Noteworthy Contributions

• The application of the Heterogeneous Multiscale Method to elliptic problems without scale separation marks a significant advancement in handling complex multiscale systems.
• The localized orthogonal decomposition method with $H^1$ interpolation provides a robust solution for multiscale elliptic problems, eliminating the need for scale separation assumptions.
• The new error analysis for finite element methods in elliptic Neumann boundary control problems broadens the applicability of these methods to both smooth and rough coefficients.
• The non-uniform $\alpha$-robust IMEX-L1 mixed finite element method for time-fractional PIDEs demonstrates strong stability and optimal error estimates, even as $\alpha$ approaches 1.
• The error-controlled cubic spline interpolation method in CNC technology addresses the challenges of maintaining $C^2$ continuity and controlling interpolation error effectively.