  Divergence-free (div-free) and curl-free vector fields are pervasive in many
areas of science and engineering, from fluid dynamics to electromagnetism. A
common problem that arises in applications is that of constructing smooth
approximants to these vector fields and/or their potentials based only on
discrete samples. Additionally, it is often necessary that the vector
approximants preserve the div-free or curl-free properties of the field to
maintain certain physical constraints. Div/curl-free radial basis functions
(RBFs) are a particularly good choice for this application as they are meshfree
and analytically satisfy the div-free or curl-free property. However, this
method can be computationally expensive due to its global nature. In this
paper, we develop a technique for bypassing this issue that combines
div/curl-free RBFs in a partition of unity framework, where one solves for
local approximants over subsets of the global samples and then blends them
together to form a div-free or curl-free global approximant. The method is
applicable to div/curl-free vector fields in $\R^2$ and tangential fields on
two-dimensional surfaces, such as the sphere, and the curl-free method can be
generalized to vector fields in $\R^d$. The method also produces an approximant
for the scalar potential of the underlying sampled field. We present error
estimates and demonstrate the effectiveness of the method on several test
problems.
