A stabilised mixed finite element method for thin plate splines based on biorthogonal systems

Bishnu Prasad Lamichhane, Markus Hegland

Abstract


We propose a novel stabilised mixed finite element method for the discretisation 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 eliminate these two variables (gradient of the smoother and Lagrange multiplier) leading to a positive definite formulation. We prove a sub-optimal a priori error estimate for the proposed finite element scheme.

References
  • A. Bab-Hadiashar, D. Suter, and R. Jarvis. Optic flow computation using interpolating thin-plate splines. In Second Asian Conference on Computer Vision (ACCV'95), pages 452--456, Singapore, 1995. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.22.2380.
  • C. Bernardi, Y. Maday, and A. T. Patera. A new nonconforming approach to domain decomposition: the mortar element method. In H. Brezzi et al., editor, Nonlinear partial differential equations and their applications, pages 13--51. Pitman, 1994.
  • J. Brandts and M. Kr\T1\i zek. Gradient superconvergence on uniform simplicial partitions of polytopes. IMA Journal of Numerical Analysis, 23:489--505, 2003. doi:10.1093/imanum/23.3.489.
  • S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Springer--Verlag, New York, 1994.
  • X. Cheng, W. Han, and H. Huang. Some mixed finite element methods for biharmonic equation. Journal of Computational and Applied Mathematics, 126:91--109, 2000. doi:10.1016/S0377-0427(99)00342-8.
  • P. G Ciarlet. The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978.
  • J. Duchon. Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics, volume 571, pages 85--100. Springer-Verlag, Berlin, 1977.
  • A. Iske. Multiresolution Methods in Scattered Data Modelling, volume 37 of LNCS. Springer, Heidelberg, 2004.
  • C. Johnson and J. Pitkaranta. Some mixed finite element methods related to reduced integration. Mathematics of Computation, 38:375--400, 1982. doi:10.1090/S0025-5718-1982-0645657-2.
  • C. Kim, R. D. Lazarov, J. E. Pasciak, and P. S. Vassilevski. Multiplier spaces for the mortar finite element method in three dimensions. SIAM Journal on Numerical Analysis, 39:519--538, 2001. doi:10.1137/S0036142900367065.
  • B. P. Lamichhane. Higher Order Mortar Finite Elements with Dual Lagrange Multiplier Spaces and Applications. LAP LAMBERT Academic Publishing, 2011.
  • B. P. Lamichhane. A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems. Journal of Computational and Applied Mathematics, 235:5188--5197, 2011. doi:10.1016/j.cam.2011.05.005.
  • B. P. Lamichhane, S. Roberts, and L. Stals. A mixed finite element discretisation of thin-plate splines. In W. McLean and A. J. Roberts, editors, Proceedings of the 15th Biennial Computational Techniques and Applications Conference, CTAC-2010, volume 52 of ANZIAM J., pages C518--C534, 2011. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3934.
  • S. Roberts, M. Hegland, and I. Altas. Approximation of a thin plate spline smoother using continuous piecewise polynomial functions. SIAM Journal on Numerical Analysis, 41:208--234, 2003. doi:10.1137/S0036142901383296.
  • D. B. Szyld. The many proofs of an identity on the norm of oblique projections. Numerical Algorithms, 42:309--323, 2006. http://link.springer.com/article/10.1007%2Fs11075-006-9046-2.
  • G. Wahba. Spline Models for Observational Data, volume 59 of Series in Applied Mathematic. SIAM, Philadelphia, first edition, 1990.
  • H. Wendland. Scattered Data Approximation. Cambridge University Press, first edition, 2005.

Keywords


Thin plate splines, scattered data smoothing, mixed finite element method, saddle point problem, biorthogonal system, a priori estimate

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DOI: http://dx.doi.org/10.21914/anziamj.v54i0.6218



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