### A mixed finite element discretisation of thin-plate splines

#### Abstract

Thin-plate splines are a well established technique for the interpolation and smoothing of scattered data. However, the traditional formulation of the method leads to large, dense and often ill-conditioned matrices, which reduces its applicability in practice. We present a new mixed finite element formulation based on the ideas behind the mortar finite element methods. The resulting system of equations is sparse and positive definite, and its size depends only on the number of finite elements not the number of data points.

**References**- http://geopubs.wr.usgs.gov/open-file/of00-043/bathymetry/appendices.html.
- C. Bernardi, Y. Maday, and A. T. Patera. A new nonconforming approach to domain decomposition: the mortar element method. In H. Brezis and J.-L. Lions, editor, {{N}onlinear Partial Differential Equations and Their Applications}, pages 13--51. Paris, 1994.
- X. Cheng, W. Han, and H. Huang. Some mixed finite element methods for biharmonic equation. {Journal of Computational and Applied Mathematics}, 126(1-2):91--109, 2000. http://dx.doi.org/10.1016/S0377-0427(99)00342-8.
- P. G. Ciarlet. {The Finite Element Method for Elliptic Problems}. North Holland, Amsterdam, 1978.
- P. G. Ciarlet and P. A. Raviart. A mixed finite element method for the biharmonic equation. In C. De Boor, editor, {Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations}, pages 125--143, New York, 1974. Academic Press.
- J. Duchon. Splines minimizing rotation-invariant semi-norms in {S}obolev spaces. In {Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics}, volume 571, pages 85--100. Springer-Verlag, Berlin, 1977. http://www.springerlink.com/content/g27671q701166031/.
- R. S. Falk. Approximation of the biharmonic equation by a mixed finite element method. {SIAM Journal on Numerical Analysis}, 15(3):556--567, 1978. http://epubs.siam.org/sinum/resource/1/sjnaam/v15/i3.
- M. F. Hutchinson. A stochastic estimator of the trace of the influence matrix for {L}aplacian smoothing splines. {Communications in Statistics --- Simulation and Computation}, 18(3):1059--1076, 1989. http://www.tandfonline.com/toc/lssp20/18/3.
- C. Johnson and J. Pitkaranta. Analysis of some mixed finite element methods related to reduced integration. {Mathematics of Computation}, 38(158):375--400, 1982. http://www.ams.org/journals/mcom/1982-38-158/home.html.
- B. P. Lamichhane, S. Roberts, and L. Stals. A mixed finite element discretization of thin plate splines based on quasi-biorthogonal systems. To be submitted.
- P. Monk. A mixed finite element method for the biharmonic equation. {SIAM Journal on Numerical Analysis}, 24(4):737--749, 1987. http://epubs.siam.org/sinum/resource/1/sjnaam/v24/i4.
- T. Ramsay. Spline smoothing over difficult regions. {Journal of Royal Statistical Society. Series B (Statistical Methodology)}, 64(2):307--319, 2002. http://onlinelibrary.wiley.com/doi/10.1111/rssb.2002.64.issue-2/issuetoc.
- S. Roberts, M. Hegland, and I. Altas. Approximation of a thin plate spline smoother using continuous piecewise polynomial functions. {SIAM Journal on Numerical Analysis}, 41(1):208--234, 2003. http://epubs.siam.org/sinum/resource/1/sjnaam/v41/i1.
- L. R. Scott and S. Zhang. Finite element interpolation of nonsmooth functions satisfying boundary conditions. {Mathematics of Computation}, 54(190):483--493, 1990. http://www.ams.org/journals/mcom/1990-54-190/home.html.
- L. Stals and S. Roberts. Smoothing large data sets using discrete thin plate splines. {Computing and Visualization in Science}, 9(3):185--195, 2006. http://www.springerlink.com/content/1432-9360/9/3/.
- G. Wahba. {Spline Models for Observational Data}, volume 59 of {Series in Applied Mathematic}. SIAM, Philadelphia, first edition, 1990.

#### Keywords

thin-plate splines, mixed finite elements

DOI: http://dx.doi.org/10.21914/anziamj.v52i0.3934

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