Both direct and iterative postprocessing methods are introduced to recover local conservation and higher order compatibility conditions, and to provide a velocity field with continuous normal components. Both methods treat a general non-conforming unstructured mesh, allow a projected mesh different from the original, handle the velocity coming from general flow schemes including non-conservative schemes, and process the velocity data from field measurements. The direct method is proved to be optimally accurate in general, whereas the iterative method consists of exponentially convergent iterations. Numerical results show that both methods not only maintain, but also improve the accuracy of the velocity.