The requirement for stabilization or other similar techniques is well known when using the finite element method in computational fluid mechanics. A variety of such techniques has been introduced during the past decades along with different physical interpretations of the stabilization terms employed. In introducing so-called information flux methods, we developed a new point of view on the problem of numerical instabilities: with respect to Shannon?s information theory instabilities are interpreted as a consequence of unadequate observance of the information flux present in fluid mechanics. Here we discuss different approaches to setting up information flux maximum-entropy approximation schemes based on that idea. By means of several numerical examples the superior accuracy of these approximation schemes is demonstrated for convection-diffusion problems compared to state-of-the-art stabilized finite element methods.