  We consider data assimilation for the heat equation using a finite element
space semi-discretization. The approach is optimization based, but the design
of regularization operators and parameters rely on techniques from the theory
of stabilized finite elements. The space semi-discretized system is shown to
admit a unique solution. Combining sharp estimates of the numerical stability
of the discrete scheme and conditional stability estimates of the ill-posed
continuous pde-model we then derive error estimates that reflect the
approximation order of the finite element space and the stability of the
continuous model. Two different data assimilation situations with different
stability properties are considered to illustrate the framework. Full detail on
how to adapt known stability estimates for the continuous model to work with
the numerical analysis framework is given in appendix.
