  Nonlinear approximations to problems with mixed boundary conditions are
useful for predicting large-scale streaming velocities from the density field,
or vice-versa. We evaluate the schemes of Bernardeau \cite{bernardeau92},
Gramann \cite{gramann93}, and Nusser \etal \cite{nusser91}, using smoothed
density and velocity fields obtained from $N$-body simulations of a CDM
universe. The approximation of Nusser \etal is overall the most accurate and
robust. For Gaussian smoothing of 1000\kms\ the mean error in the approximated
relative density perturbation, $\delta$, is smaller than 0.06, and the
dispersion is 0.1. The \rms\ error in the estimated velocity is smaller than
60\kms, and the dispersion is 40\kms. For smoothing of 500\kms\ these numbers
increase by about a factor $\sim 2$ for $\delta < 4-5$, but deteriorate at
higher densities. The other approximations are comparable to those of Nusser
\etal for smoothing of 1000\kms, but are much less successful for the smaller
smoothing of 500\kms.
