The authors study a finite difference method for approximating the unknown source parameter and of the following inverse problem: find and which satisfy (1) in ; , ; on ; subject to the additional specification (2) , , , where , , , , , and are known functions, and is a fixed prescribed interior point in whose boundary is denoted by . If represents the temperature then the problem can be viewed as a control problem of finding the control such that the internal constraint (2) is satisfied.
The backward Euler scheme is studied and its convergence is proved via an application of the discrete maximum principle for a transformed problem. The approximation of and in terms of the approximation obtained for the transformed problem is discussed. Finally the paper contains some numerical computations for several examples which support the theoretical analysis.