Numerical Analysis of Space-Time Dependent Source Identification in Subdiffusion Equations

TL;DR

This paper introduces a fixed-point algorithm for reconstructing space-time dependent sources in subdiffusion equations, combining finite element and finite difference methods, with proven convergence and error bounds.

Contribution

The work presents a novel, easy-to-implement fixed-point approach with rigorous convergence analysis for inverse source problems in subdiffusion models.

Findings

The fixed-point iteration converges linearly.

Error bounds depend explicitly on discretization and noise.

Numerical experiments confirm theoretical results.

Abstract

In this work, we propose an easy-to-implement fixed-point algorithm for reconstructing a space-time dependent source in a subdiffusion model from lateral boundary measurements. The numerical scheme combines a Galerkin finite element method for spatial discretization with a finite difference method for temporal discretization. We establish the linear convergence of the fixed-point iteration and derive an error bound that depends explicitly on the discretization parameters and the noise level. The error analysis relies on stability properties of the continuous inverse problem and technical estimates for the associated direct problem with limited-regularity data. Numerical experiments are presented to support and complement the theoretical analysis.