Convergence and error estimates for pseudo-polyharmonic div-curl and elastic interpolation on a bounded domain




Approximation theory , interpolation and approximation, convergence and error estimates, numerical analysis, functional analysis
Abstract views: 203


This paper establishes convergence rates and error estimates for the pseudo-polyharmonic div-curl and elastic interpolation. This type of interpolation is based on a combination of the divergence and the curl of a multivariate vector field and minimizing an appropriate functional energy related to the divergence and curl. Convergence rates and error estimates are established when the interpolated vector field is assumed to be in the classical fractional vectorial Sobolev space on an open bounded set with a Lipschitz-continuous boundary. The error estimates introduced in this work are sharp and the rate of convergence depends algebraically on the fill distance of the scattered data nodes. More precisely, the order of convergence depends, essentially, on the smoothness of the target vector field, on the dimension of the Euclidean space and on the null space of corresponding Sobolev semi-norm.


Download data is not yet available.


R.A. Adams, Sobolev spaces, Academic Press, New York, 1975.

R.A. Adams, J.J.F. Fournier, Sobolev spaces, 2nd ed, Academic Press, New York, 2003.

L. Amodei, M.N. Benbourhim, A vector spline approximation, J. Approx. Theory, 67 (1991), pp. 51–79. DOI:

R. Arcangeli, M.C. Lopez de Silanes, J.J. Torrens, Multidimensional Minimizing Spline functions, Kluwer Academic Publishers, 2004. DOI:

R. Arcangeli, M.C. Lopez de Silanes, J.J. Torrens, An extension of a bound for functions in Sobolev spaces, with applications to (m, s)-spline interpolation and smoothing, Numer. Math., 107 (2007), pp. 181–211. DOI:

M. Atteia, Hilbertian Kernels and spline functions, North-Holland, Elsevier Science, 1992.

M.N. Benbourhim, A. Bouhamidi, Approximation of vector fields by thin plate splines with tension, J. Approx. Theory, 136 (2005), pp. 198–229. DOI:

M.N. Benbourhim, A. Bouhamidi, Pseudo-polyharmonic vectorial approximation for div-curl and elastic semi-norms, Numer. Math., 109 (2008), pp. 333–364. DOI:

M.N. Benbourhim, A. Bouhamidi, Error estimates for interpolating div-curl splines under tension on a bounded domain, J. Approx. Theory, 152 (2008), pp. 68–81. DOI:

S. Bergman, Schiffer, Kernel functions and elliptic differential equations in mathematical physics, Academic Press, 1953.

A.Y. Bezhaev, V.A. Vasilenko, Variational Theory of Splines, New York, Kluwer Academic, Plenum Publishers, 2001. DOI:

M.D. Buhmann, Radial basis functions: Theory and implementation, Cambridge monographs on Applied Computational Mathematics, 12, Cambridge University Press, 2003.

F. Chen, S. Suter, Elastic spline models for human cardiac motion estimation, IEEE Nonrigid and Articulated Motion Workshop, Puerto Rico, 1997, pp. 120–127, DOI:

F. Chen, S. Suter, Image coordinate transformation based on div-curl vector splines, in: Proc. 14th Int. Conf. Pattern Recognition, (ICPR’98), I, Brisbane, Australia, pp. 518–520, Aug. 16–20, 1998. DOI:

P. Ciarlet, The Finite Element Method for Elliptic Problems, Amsterdam, North-Holland, 1978. DOI:

F. Dodu, C. Rabut, Vectorial interpolation using radial-basis-like functions, Comput. Math. Appl., 43 (2002), pp. 393–411. DOI:

J. Duchon, Splines minimizing translation-invariant seminorms in Sobolev spaces, in: Constructive Theory of Functions of Several Variables, eds. W. Schempp and K. Zeller, Lecture Notes in Mathematics, 571, Springer-Verlag, Berlin, 1977, pp. 85–100, DOI:

J. Duchon, Sur l’erreur d’interpolation des fonctions de plusieurs variables par les Dm-spline, RAIRO Analyse Numerique, 12 (1978) no. 4., pp. 325–334. DOI:

T. Iwaniec, C. Sbordone, Quasiharmonic fields, Annales de l’Institut Henri Poincare, 18 (2001) no. 5, pp. 519–572, DOI:

J. Hinkle, M. Szegedi, B. Wang, B. Salter, S. Joshi, 4D CT image reconstruction with diffeomorphic motion model, Medical Image Analysis, 16 (2012) no. 6, pp. 1307– 1316, DOI:

P.J. Laurent, Approximation et optimisation, Hermann, Paris, 1972.

W.R. Madych, An estimate for multivariate interpolation II, J. Approx. Theory, 142 (2006), pp. 116–128. DOI:

W.R. Madych, E.H. Potter, An estimate for multivariate interpolation, J. Approx. Theory, 43 (1985), pp. 132–139, DOI:

F.J. Narcowich, J.D. Ward, H. Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Math. Comput., 74 (2005), pp. 743–763, DOI:

F.J. Narcowich, J.D. Ward, H. Wendland, Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions, Constructive Approximation, 24 (2006), pp. 175–186. DOI:

J. Necas, Les Methodes Directes en Theorie des Equations Elliptiques, Masson, Paris, 1967.

J. Peetre, Espaces d’interpolation et theoreme de Soboleff, Ann. Inst. Fourier, Grenoble, 16 (1966), pp. 279–317. DOI: DOI:

A.M. Sanchez, R. Arcangeli, Estimations des erreurs de meilleure approximation polynomiale et d’interpolation de Lagrange dans les espaces de Sobolev d’ordre non entier. Numer. Math., 45 (1984), pp. 301–321, DOI:

L. Schwartz, Theorie des distibutions, Hermann, Paris, 1966.

F. Ong, M. Uecker, U. Tariq, A. Hsiao, M.T. Alley, S.S. Vasanawala, M.Lustig, Robust 4D flow denoising using divergence-free wavelet transform, Magnetic Resonance in Medicine, 73 (2015) no. 2, pp 1522–2594. DOI:

E. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, 1970. DOI:

D. Suter, Motion estimation and vector splines, in: Proc. CVPR’94, Seattle WA, IEEE, Institute of Electrical and Electronics Engineers, (1994), pp. 939–942. DOI:

D. Suter, F. Chen, Left ventricular motion reconstruction based on elastic vector splines, IEEE Transactions on Medical Imaging, 19 (2000) no. 4„ pp. 295–305, DOI:

C.O.S. Sorzano, P. Thevenaz, M. Unser, Elastic registration of biological images using vector-splines regularization, IEEE Trans. Biomedical Engineering, 52 (2005) no. 4, pp. 652–663, DOI:

D. Trad, Five-dimensional interpolation: Recovering from acquisition constraints, Geo physics, 74 (2009) no. 6, pp. 123–132, DOI:

D. Trad, Five Dimensional Seismic Data Interpolation, Frontiers+Innovations, CSPG, CSEG, CWLS, Conventions, Calgary, Alberta, Canada, (2009), pp. 689–692.

K. Watanabe, K. Matsuno, Moving computational domain method and its application to flow around a high-speed car passing through a hairpin curve, Journal of Computational Science and Technology, 3 (2009) no. 2, pp. 449–459, DOI:




How to Cite

Benbourhim, M.-N., Bouhamidi, A., & Gonzalez-Casanova, P. (2023). Convergence and error estimates for pseudo-polyharmonic div-curl and elastic interpolation on a bounded domain. J. Numer. Anal. Approx. Theory, 52(1), 34–56.