Discrete maximal regularity of time-stepping schemes for fractional evolution equations

Area of Science:

• Numerical Analysis
• Fractional Calculus
• Partial Differential Equations

Background:

• Fractional evolution models involve fractional derivatives in time, presenting unique analytical challenges.
• Understanding the regularity of numerical solutions is crucial for stability and accuracy.
• Existing methods for parabolic problems need extension to fractional settings.

Purpose of the Study:

• To establish maximal [Formula: see text]-regularity for various time stepping schemes applied to fractional evolution models.
• To analyze the performance of specific numerical methods including convolution quadratures, L1 scheme, and fractional Crank-Nicolson.
• To generalize existing regularity results from parabolic problems to fractional evolution equations.

Main Methods:

• Application of operator-valued Fourier multiplier theorems (Weis, Blunck) for theoretical analysis.
• Analysis of convolution quadratures generated by backward Euler and second-order backward difference formulas.
• Investigation of the L1 scheme, explicit Euler method, and a fractional Crank-Nicolson variant.

Main Results:

• Maximal [Formula: see text]-regularity is established for the considered time stepping schemes.
• The analysis provides a rigorous framework for assessing the regularity of numerical solutions for fractional models.
• The findings demonstrate the applicability and effectiveness of these schemes in the context of fractional calculus.

Conclusions:

• The established maximal [Formula: see text]-regularity results offer significant advancements in the numerical analysis of fractional evolution equations.
• These findings generalize and extend previous results known for parabolic problems.
• The study provides a robust theoretical foundation for the numerical simulation of fractional dynamics.