Interpolating Solenoidal Vector Field From Irregular Data Points

Question

Assume you measure the flow in a body of water (or some higher dimensional equivalent) $U\subseteq\mathbf{R}^n$ in $k$ points (not necessarily on some grid), so you get a data set $\{(x_i,y_i)\}_{i<k}$ with $(x_i,y_i)\in U\times\mathbf{R}^n$. What techniques exist to interpolate from those data points to a solenoidal vector field $f:U\to\mathbf{R}^n$ such that $f(x_i)=y_i$ for $i<k$? Where $U$ is possibly given by some polytype?