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

Abstract

We investigate the error of the (semidiscrete) Galerkin method applied to a semilinear subdiffusion equation in the presence of nonsmooth initial data. The diffusion coefficient is allowed to depend on time. We use the energy method to find optimal bounds on the error under weak natural assumptions on the diffusivity. First, we prove the result for the linear problem and then use the “frozen nonlinearity” technique coupled with various generalizations of Grönwall inequality to carry the result to the semilinear case. The paper ends with numerical illustrations supporting the theoretical results.