The authors consider an inverse problem of an abstract elliptic differential equation with respect to a general Banach space. The problem is overdetermined and thus an operator-valued source term has to be identified such that a solution exists.
The authors investigate two scenarios. Firstly, a general semidiscretisation yields a sequence of abstract ordinary differential equations (ODEs) of second order. Secondly, the abstract ODEs are discretised by the symmetric difference quotient of second order. The convergence of the approximations is analysed in corresponding Banach spaces, where operator theory as well as the theory of analytic -semigroups are applied. Assuming uniformly positive operators in the semidiscretisation, the authors prove the uniform convergence in both scenarios with a constant operator as source term.
Furthermore, the authors investigate the case of a class of time-dependent operators as source term. Again the uniform convergence is proved in both scenarios under certain assumptions. The discussion of test examples or numerical simulations are not within the scope of the paper.