Error estimates for the full discretization of a semilinear parabolic problem with an unknown source

This paper is devoted to the study of an inverse semilinear parabolic problem. The problem contains an unknown solely time-dependent source function p and a homogeneous Dirichlet boundary condition. Moreover, an integral measurement of the total energy/mass in the domain is given. A full-discrete finite element scheme to approximate the unique weak solution is designed. For the time discretization backward Euler's method is used. For the space discretization the finite element method is applied. Various error estimates are derived, depending on the regularity of the data and on the choice of the finite elements.