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We investigate the error of the (semidiscrete) Galerkin method applied to asemilinear subdiffusion equation in the presence of a nonsmooth initial data.The diffusion coefficient is allowed to depend on time. It is well-known thatin such parabolic problems the spatial error increases when time decreases. Therate of this time-dependency is a function of the fractional parameter and theregularity of the initial condition. We use the energy method to find optimalbounds on the error under weak and natural assumptions on the diffusivity.First, we prove the result for the linear problem and then use the "frozennonlinearity" technique coupled with various generalizations of Gronwallinequality to carry the result to the semilinear case. The paper ends withnumerical illustrations supporting the theoretical results.

Error of the Galerkin scheme for a semilinear subdiffusion equation with time-dependent coefficients and nonsmooth data