Numerically calculating the divergence of a set of oriented points

Question

Say I have a set of oriented points at locations $\vec{v_i}$ with each some direction $\vec{n_i}$, in practice they represent a surface with its normals. How would I calculate the divergence of this vector field? Standard finite-difference methods would obviously not work because there is no grid.

Can I perhaps assume a grid and use nearest neighbours to infer the vector field at each point? The discretized vector field would then have at each point a weighted average of nearby points. This would then allow me to use standard FD-methods to calculate the divergence at each gridpoint. Is this a common method, or are there better methods?

Answer 1

Interpolation of vector fields (a notoriously tricky process) to determine the divergence can be avoided by using a line integral definition of the divergence via Stokes' theorem. You can find a discussion of this method in On the interpolation of a vector field (1979). For a more recent discussion see A Quantitative Comparison between Traditional and Line Integral Methods of Derivative Estimation (2001).

The “traditional” approach to estimating vector field derivatives extracts a gridded field from the original observations, which are typically not uniformly distributed in space. However, there exist other methods involving derivative estimation via line integral (“triangle”) techniques that do not involve a prior mapping of the field onto a uniform grid. Empirical testing of the differences between vector field derivative estimation using two different schemes is done with prototypical examples of the techniques in the context of meteorological wind data. The test results make it clear that the traditional method is generally inferior to derivative estimates via the line integral methodology.