  A material-based, i.e., Lagrangian, methodology for exact integration of flux
by volume-preserving flows through a surface has been developed recently in
[Karrasch, SIAM J. Appl. Math., 76 (2016), pp. 1178-1190]. In the present
paper, we first generalize this framework to general compressible flows,
thereby solving the donating region problem in full generality. Second, we
demonstrate the efficacy of this approach on a slightly idealized version of a
classic two-dimensional mixing problem: transport in a cross-channel
micromixer, as considered recently in [Balasuriya, SIAM J. Appl. Dyn. Syst., 16
(2017), pp. 1015-1044].
