Multilevel interpolation of divergence-free vector fields

We introduce a multilevel technique for interpolating scattered data of divergence-free vector fields with the help of matrix-valued compactly supported kernels. The support radius at a given level is linked to the mesh norm of the data set at that level. There are at least three advantages of this method: no grid structure is necessary for the implementation, the multilevel approach is computationally cheaper than solving a large one-shot system and the interpolant is guaranteed to be analytically divergence-free. Furthermore, though we will not pursue this here, our multilevel approach is able to represent multiple scales in the data if present. We will prove convergence of the scheme, stability estimates and give a numerical example. For the first time, we will also prove error estimates for derivatives and give approximation orders in terms of the fill distance of the finest data set.