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

TL;DR

This paper analyzes the error bounds of the Galerkin method for semilinear subdiffusion equations with time-dependent coefficients and nonsmooth data, providing optimal estimates and numerical validation.

Contribution

It introduces a novel error analysis framework for semilinear subdiffusion equations with time-dependent coefficients and nonsmooth initial data, extending existing methods.

Findings

Optimal error bounds are established under weak assumptions.

The analysis covers both linear and semilinear cases.

Numerical results confirm the theoretical error estimates.

Abstract

We investigate the error of the (semidiscrete) Galerkin method applied to a semilinear subdiffusion equation in the presence of a nonsmooth initial data. The diffusion coefficient is allowed to depend on time. It is well-known that in such parabolic problems the spatial error increases when time decreases. The rate of this time-dependency is a function of the fractional parameter and the regularity of the initial condition. We use the energy method to find optimal bounds on the error under weak and 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 Gronwall inequality to carry the result to the semilinear case. The paper ends with numerical illustrations supporting the theoretical results.