# Finite element approximation of source term identification with TV-regularization

## TL;DR

This paper develops a finite element approach combined with TV-regularization to accurately recover discontinuous source terms in elliptic systems from boundary measurements, ensuring stability and convergence.

## Contribution

It introduces a finite element discretization for TV-regularized source identification and proves the convergence and stability of the proposed algorithm.

## Key findings

- The method is stable and convergent.
- Numerical experiments validate theoretical results.

## Abstract

In this paper we investigate the problem of recovering the source term in an elliptic system from a measurement of the state on a part of the boundary. For the particular interest in reconstructing probably discontinuous sources, we use the standard least squares method with the total variation regularization. The finite element method is then applied to discretize the minimization problem, we show the stability and the convergence of this technique. Furthermore, we have proposed an algorithm to stably solve the minimization problem. We prove the iterate sequence generated by the derived algorithm converging to a minimizer of the regularization problem, and that convergence measurement is also established. Finally, a numerical experiment is presented to illustrate our theoretical findings.