  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.
