In the finite cell method (FCM), the computational domain is embedded in an extended, in general simply shaped domain that can be easily meshed, e.g. by a regular grid of square or cube-shaped elements. The original domain is retained by defining an indicator function alpha, which is 1 in its interior and attains a small value alpha_zero in the fictitious part outside. Formally, this indicator function is multiplied to suitable material parameters in a weak formulation of the PDE to be solved. For a model problem in structural mechanics, alpha thus adds material of very small stiffness in the exterior part. Elements that are partially inside and partially outside the domain are called ‘cut cells’ and require special consideration due to the discontinuity in their element integrals. This basic concept of FCM (and that of very similar other immersed boundary methods) shows that several types of approximation errors must be considered and, if possible, be controlled: The definition of a non-zero value alpha_zero in the fictitious part of the computational domain introduces a modeling error through a ‘fictitious stiffness’. In cut cells the integration error must be controlled and, last but not least, the influence of the embedded approach on the approximation properties of the underlying finite element computation, in particular in the context of mesh refinement and degree adaptation needs special consideration. This presentation discusses these error types and describes concepts for their control, based on a priori knowledge and/or a posteriori computation. Finally, we will extend this to an inverse problem, where the identification of the model itself is in the focus of interest.