*(English)*Zbl 0805.65133

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

##### MSC:

65Z05 | Applications of numerical analysis to physics |

65M30 | Improperly posed problems (IVP of PDE, numerical methods) |

65M06 | Finite difference methods (IVP of PDE) |

35K15 | Second order parabolic equations, initial value problems |

35R30 | Inverse problems for PDE |