Journal of Scientific Computing

, Volume 67, Issue 1, pp 20–42 | Cite as

A Mixed Finite Element Discretisation of Thin Plate Splines Based on Biorthogonal Systems

  • Bishnu P. Lamichhane
  • Stephen G. Roberts
  • Linda Stals


The thin plate spline method is a widely used data fitting technique as it has the ability to smooth noisy data. Here we consider a mixed finite element discretisation of the thin plate spline. By using mixed finite elements the formulation can be defined in-terms of relatively simple stencils, thus resulting in a system that is sparse and whose size only depends linearly on the number of finite element nodes. The mixed formulation is obtained by introducing the gradient of the corresponding function as an additional unknown. The novel approach taken in this paper is to work with a pair of bases for the gradient and the Lagrange multiplier forming a biorthogonal system thus ensuring that the scheme is numerically efficient, and the formulation is stable. Some numerical results are presented to demonstrate the performance of our approach. A preconditioned conjugate gradient method is an efficient solver for the arising linear system of equations.


Thin plate splines Scattered data smoothing Mixed finite element method Biorthogonal system 

Mathematics Subject Classification

65D15 41A15 



We are grateful to the anonymous referees for their valuable suggestions to improve the quality of the earlier version of this work.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Bishnu P. Lamichhane
    • 1
  • Stephen G. Roberts
    • 2
  • Linda Stals
    • 2
  1. 1.School of Mathematical & Physical Sciences, Mathematics Building - V127University of Newcastle, University DriveCallaghanAustralia
  2. 2.Centre for Mathematics and its Applications, Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia

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