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

TL;DR

This paper introduces a new stabilized mixed finite element method for thin plate splines that uses biorthogonal systems to achieve a positive definite formulation and optimal error estimates.

Contribution

It presents a novel stabilized mixed finite element approach for thin plate splines utilizing biorthogonal systems for variable elimination and improved stability.

Findings

Achieves a positive definite formulation for thin plate splines

Provides optimal a priori error estimates

Utilizes biorthogonal systems for efficient variable elimination

Abstract

The thin plate spline is a popular tool for the interpolation and smoothing of scattered data. In this paper we propose a novel stabilized mixed finite element method for the discretization 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 can easily eliminate these two variables (gradient of the smoother and Lagrange multiplier) leading to a positive definite formulation. The optimal a priori estimate is proved by using a superconvergence property of a gradient recovery operator.