Divergence-free and curl-free moving least squares approximations

This paper presents a vector-valued moving least squares (MLS) approximation for reconstructing vector fields that are divergence-free or curl-free. The proposed method constructs analytically divergence-free or curl-free shape functions by applying appropriate differential operators to a stream (potential) function approximated using the MLS method. The procedure involves solving a sparse linear least squares problem, for which the Conjugate Gradient Least Squares (CGLS) algorithm is employed to reduce computational costs compared to the direct solvers. The approach relies solely on a set of scattered nodes in the computational domain and requires no background triangulation. We provide error bounds for the approximation and support the theoretical bounds with numerical experiments. Additionally, we demonstrate the application of the divergence-free MLS approximation to the numerical solution of Darcy's flow equations.