A Mixed Finite Element Discretisation of Thin Plate Splines Based on Biorthogonal Systems
- 189 Downloads
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.
KeywordsThin plate splines Scattered data smoothing Mixed finite element method Biorthogonal system
Mathematics Subject Classification65D15 41A15
We are grateful to the anonymous referees for their valuable suggestions to improve the quality of the earlier version of this work.
- 2.Altas, I., Hegland, M., Roberts, S.: Finite element thin plate splines for surface fitting. In: Computational Techniques and Applications: CTAC97, pp. 289–296 (1998)Google Scholar
- 3.Arnold, D., Brezzi, F.: Some new elements for the Reissner-Mindlin plate model. In: Boundary Value Problems for Partial Differerntial Equations and Applications, pp. 287–292. Masson, Paris (1993)Google Scholar
- 9.Ciarlet, P.: The finite element method for elliptic problems. North Holland, Amsterdam (1978)Google Scholar
- 10.Ciarlet, P., Raviart, P.-A.: A mixed finite element method for the biharmonic equation. In: Boor, C.D. (ed.) Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 125–143. Academic Press, New York (1974)Google Scholar
- 11.Duchon, J.: Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In: Constructive Theory of Functions of Several Variables. Lecture Notes in Mathematics, vol. 571. Springer-Verlag, Berlin (1977)Google Scholar
- 19.Lamichhane, B.: Higher Order Mortar Finite Elements with Dual Lagrange Multiplier Spaces and Applications. LAP LAMBERT Academic Publishing (2011)Google Scholar
- 22.Lamichhane, B., Hegland, M.: A stabilised mixed finite element method for thin plate splines based on biorthogonal systems. In: McLean, W., Roberts, A.J. (eds.) Proceedings of the 16th Biennial Computational Techniques and Applications Conference, CTAC-2012, ANZIAM J. (2013)Google Scholar
- 23.Lamichhane, B., Roberts, S., Stals, L.: A mixed finite element discretisation of thin-plate splines. In: McLean, W., Roberts, A.J. (eds.), Proceedings of the 15th Biennial Computational Techniques and Applications Conference, CTAC-2010, vol. 52 of ANZIAM J, pp. C518–C534 (2011)Google Scholar
- 28.Wahba, G.: Spline Models for Observational Data, vol. 59 of Series in Applied Mathematic, SIAM, Philadelphia, first ed., (1990)Google Scholar