Spline approximation of gradient fields: Applications to wind velocity fields

We study a spline-based approximation of vector fields in the conservative case. This problem appears for instance when approximating current or wind velocity fields, the data deriving in those cases from a potential (pressure for the wind, etc..). In the modelling, we introduce a minimization problem on an Hilbert space for which the existence and uniqueness of the solution are provided. A convergence result in the introduced Sobolev space is established using norm equivalence and compactness arguments, as well as an approximation error estimate of the involved smoothing D-splines.