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We study a vectorial approximation problem based on thin plate splines with tension involving two positive parameters: one for the control of the oscillations and the other for the control of the divergence and rotational components of the field. The existence and uniqueness of the solution are proved and the solution is explicitly given. As special cases, we study the limit problems as the parameter controlling the divergence and the rotation converges to zero or infinity. The divergencefree and the rotation-free approximation problems are also considered. The convergence in Sobolev space is studied.

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Vector field reconstruction is a problem arising in many scientific applications. In this paper, we study a div-curl approximation of vector fields by pseudo-polyharmonic splines. This leads to the variational smoothing and interpolating spline problems with minimization of an energy involving the curl and the divergence of the vector field. The relationship between the div-curl energy and elastic energy is established. Some examples are given to illustrate the effectiveness of our approach for a vector field reconstruction. Mathematics Subject Classification (2000)41A15 · 65D05 · 65D10 · 46F10