We derive optimal \(L^2\)-error estimates for semilinear time-fractional subdiffusion problems involving Caputo derivatives in time of order \(\alpha \in (0,1)\), for cases with smooth and nonsmooth initial data. A general framework is introduced allowing a unified error analysis of Galerkin type space approximation methods. The analysis is based on a semigroup type approach and exploits the properties of the inverse of the associated elliptic operator. Completely discrete schemes are analyzed in the same framework using a backward Euler convolution quadrature method in time. Numerical examples including conforming, nonconforming and mixed finite element methods are presented to illustrate the theoretical results.

对于具有平滑和非平滑初始数据的情况，我们针对涉及Caputo导数的半线性时间分数次扩散问题，以时间\（\ alpha \ in（0,1）\）的时间得出最优\（L ^ 2 \） -误差估计。介绍了一个通用框架，该框架允许对Galerkin类型空间逼近方法进行统一的误差分析。该分析基于半群类型方法，并利用了相关的椭圆算子的逆的性质。在同一框架中，使用后向Euler卷积正交方法及时分析了完全离散的方案。数值算例包括相容，非相容和混合有限元方法，以说明理论结果。