# Convergence analysis of a Crank-Nicolson Galerkin method for an inverse source problem for parabolic equations with boundary observations

## TL;DR

This paper analyzes the convergence of a Crank-Nicolson Galerkin method for an inverse source problem in parabolic equations, providing theoretical proofs and numerical validation for the method's accuracy and stability.

## Contribution

It introduces a convergence analysis for a novel Crank-Nicolson Galerkin approach to inverse source problems with boundary data, including error bounds and rates.

## Key findings

- Proved convergence of the regularized approximations as noise and mesh size decrease.
- Established error bounds and convergence rates under source conditions.
- Numerical experiments confirm theoretical results.

## Abstract

This work is devoted to an inverse problem of identifying a source term depending on both spatial and time variables in a parabolic equation from single Cauchy data on a part of the boundary. A Crank-Nicolson Galerkin method is applied to the least squares functional with an quadratic stabilizing penalty term. The convergence of finite dimensional regularized approximations to the sought source as measurement noise levels and mesh sizes approach to zero with an appropriate regularization parameter is proved. Moreover, under a suitable source condition, an error bound and corresponding convergence rates are proved. Finally, several numerical experiments are presented to illustrate the theoretical findings.