Discrete maximal regularity of time-stepping schemes for fractional evolution equations
Provides theoretical foundation for numerical analysis of fractional evolution equations, which is incremental as it extends known parabolic results.
Established maximal ℓ^p-regularity for several time-stepping schemes for fractional evolution equations with fractional derivative order α∈(0,2), α≠1, generalizing results for parabolic problems.
In this work, we establish the maximal $\ell^p$-regularity for several time stepping schemes for a fractional evolution model, which involves a fractional derivative of order $α\in(0,2)$, $α\neq 1$, in time. These schemes include convolution quadratures generated by backward Euler method and second-order backward difference formula, the L1 scheme, explicit Euler method and a fractional variant of the Crank-Nicolson method. The main tools for the analysis include operator-valued Fourier multiplier theorem due to Weis [48] and its discrete analogue due to Blunck [10]. These results generalize the corresponding results for parabolic problems.