The thin plate spline method is a widely used data fitting technique which has the ability to smooth noisy data. We present some example applications of a new mixed finite element discretisation of the thin plate spline method. The new approach works 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. We overview of the theoretical foundations of the new approach and give numerical examples in both two and three dimensions. References D. N. Arnold and F. Brezzi. Some new elements for the Reissner–Mindlin plate model. In Boundary Value Problems for Partial Differential Equations and Applications, pages 287–292. Masson, Paris, 1993. D. Boffi and C. Lovadina. Analysis of new augmented Lagrangian formulations for mixed finite element schemes. Numer. Math., 75:405–419, 1997. doi:10.1007/s002110050246 X.-L. Cheng, W. Han, and H.-C. Huang. Some mixed finite element methods for biharmonic equation. J. Comput. Appl. Math., 126:91–109, 2000. doi:10.1016/S0377-0427(99)00342-8 J. Duchon. Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics, 571:85–100. Springer-Verlag, Berlin, 1977. doi:10.1007/BFb0086566 V. Girault and P.-A. Raviart. Finite Element Methods for Navier–Stokes Equations. Springer-Verlag, Berlin, 1986. doi:10.1007/978-3-642-61623-5 M. F. Hutchinson. A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines. Commun. Stat. Simulat. Comput., 19:433–450, 1990. doi:10.1080/03610919008812866 C. Johnson and J. Pitkaranta. Analysis of some mixed finite element methods related to reduced integration. Math. Comput., 38(158):375–400, 1982. doi:10.2307/2007276 T. Karper, K.-A. Mardal, and R. Winther. Unified finite element discretizations of coupled Darcy–Stokes flow. Numer. Meth. Part. D. E., 25:311–326, 2009. doi:10.1002/num.20349 B. P. Lamichhane. A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems. J. Comput. Appl. Math., 235:5188–5197, 2011. doi:10.1016/j.cam.2011.05.005 B. P. Lamichhane, S. G. Roberts, and L. Stals. A mixed finite element discretisation of thin-plate splines. In W. McLean and A. J. Roberts (Eds), Proceedings of the 15th Biennial Computational Techniques and Applications Conference, CTAC-2010, ANZIAM J., 52:C518–C534, 2010. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3934 B. P. Lamichhane, S. G. Roberts, and L. Stals. A mixed finite element discretisation of thin plate splines based on biorthogonal systems. J. Sci. Comput., 1–23, July 2015. doi:10.1007/s10915-015-0068-6 G. Wahba. Spline Models for Observational Data, volume 59 of Series in Applied Mathematic. SIAM, Philadelphia, 1990. doi:10.1137/1.9781611970128
@article{9368,
title = {Applications of a finite element discretisation of thin plate splines},
journal = {ANZIAM Journal},
volume = {56},
year = {2016},
doi = {10.21914/anziamj.v56i0.9368},
language = {EN},
url = {http://dml.mathdoc.fr/item/9368}
}
Stals, Linda; Lamichhane, Bishnu. Applications of a finite element discretisation of thin plate splines. ANZIAM Journal, Tome 56 (2016) . doi : 10.21914/anziamj.v56i0.9368. http://gdmltest.u-ga.fr/item/9368/