A new smoothing method is proposed which can be viewed as a finite element thin plate spline. This approach combines the favorable properties of finite element surface fitting with those of thin plate splines. The method is based on first order techniques similar to mixed finite element techniques for the biharmonic equation. The existence of a solution to our smoothing problem is demonstrated, and the approximation theory for uniformly spread data is presented in the case of both exact and noisy data. This convergence analysis seems to be the first for a discrete smoothing spline with data perturbed by white noise. Numerical results are presented which verify our theoretical results and demonstrate our method on a large real life data set.