Galerkin Type Methods for Semilinear Time-Fractional Diffusion Problems

We derive optimal L²-error estimates for semilinear time-fractional subdiffusion problems involving Caputo derivatives in time of order 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.