The authors study a finite difference method for approximating the unknown source parameter $p=p(t)$ and $u=u(x,y,t)$ of the following inverse problem: find $u=u(x,y,t)$ and $p=p(t)$ which satisfy (1) $u\sb t=\Delta u+p(t)u+f(x,y,t)$ in $Q\sb T$; $u(x,y,0)=\phi(x,y)$, $(x,y)\in\Omega$; $u(x,y,t)=g(x,y,t)$ on $\partial\Omega\times[0,T]$; subject to the additional specification (2) $u(x\sp*,y\sp*,t)=E(t)$, $(x\sp*,y\sp*)\in\Omega$, $0\le t\le T$, where $Q\sb T=\Omega\times(0,T]$, $T>0$, $\Omega=(0,1)\times(0,1)$, $f$, $\phi$, $g$ and $E\ne 0$ are known functions, and $(x\sp*,y\sp*)$ is a fixed prescribed interior point in $\Omega$ whose boundary is denoted by $\partial\Omega$. If $u$ represents the temperature then the problem $(1)-(2)$ can be viewed as a control problem of finding the control $p=p(t)$ 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 $u$ and $p$ 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.