LAUSR

We derive optimal $$L^2$$ L 2 -error estimates for semilinear time-fractional subdiffusion problems involving Caputo derivatives in time of order $$\alpha \in (0,1)$$ α ∈ ( 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.