A Stabilised Mixed Finite Element Method for Thin Plate Splines Based on Biorthogonal Systems

Abstract

We propose a novel stabilised mixed finite element method for the discretisation of thin plate splines. The mixed formulation is obtained by introducing the gradient of the smoother as an additional unknown. Working with a pair of bases for the gradient of the smoother and the Lagrange multiplier, which forms a biorthogonal system, we eliminate these two variables (gradient of the smoother and Lagrange multiplier) leading to a positive definite formulation. We prove a sub-optimal a priori error estimate for the proposed finite element scheme.