Verifying convergence rates of discrete thin-plate splines in 3D

Traditional thin-plate splines use radial basis functions that produce dense linear system of equations whose size increases with the number of data points. We present a discrete thin-plate spline method that uses polynomials with local support defined on finite-element grids. The resulting system of equations is sparse and its size depends only on the number of nodes in the finite element grid. Theory is developed for general d-dimensional data sets and model problems are presented in 3D to study the convergence behaviour.