Fully discrete Galerkin scheme for a semilinear subdiffusion equation with nonsmooth data and time-dependent coefficient

TL;DR

This paper introduces a stable and convergent fully discrete Galerkin scheme for semilinear subdiffusion equations with nonsmooth data and time-dependent coefficients, validated through theoretical analysis and numerical experiments.

Contribution

It presents a novel linear numerical method coupling L1 discretization with Galerkin schemes for complex subdiffusion problems with minimal regularity assumptions.

Findings

Optimal pointwise in space error estimates

Global in time convergence results

Numerical experiments confirming theoretical predictions

Abstract

We couple the L1 discretization of the Caputo fractional derivative in time with the Galerkin scheme to devise a linear numerical method for the semilinear subdiffusion equation. Two important points that we make are: nonsmooth initial data and time-dependent diffusion coefficient. We prove the stability and convergence of the method under weak assumptions concerning regularity of the diffusivity. We find optimal pointwise in space and global in time errors, which are verified with several numerical experiments.