In this paper, we propose a method belonging to the class of special meshless kernel techniques for solving Poisson problems. The method is based on a simple new construction of an approximate particular solution of the nonho-mogeneous equation in the space of polyharmonic splines. High order Lagrangian polyharmonic splines are used as a basis to approximate the nonhomogeneous term and a closed-form particular solution is given. The coefficients can be computed by convolution products of known vectors. This can be done in all dimensions, without numerical integration nor solution of systems. Numerical experiments are presented in two dimensions and show the quality of the approximations for different test examples.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bacchelli, B., M. Bozzini, and C. Rabut, Polyharmonic wavelets based on Lagrangean functions, in Curve and Surface Fitting: Avignon 2006, A. Cohen, J.L. Merrien, and L.L. Schumaker (eds.), Nashboro Press, Brentwood, TN, 2007, 11–20
Buhmann, M.D., Multivariate cardinal interpolation with radial basis functions, Constr.Approx. 6 (1990) 225–255
Chen, C.S., G. Kuhn, J. Ling, and G. Mishuris, Radial basis functions for solving near singular Poisson problems, Commun. Numer. Meth. Eng. 19 (2003), 333–347
Hörmander, L., The Analysis of Linear Partial Differential Operators, Springer, Berlin, Vol. 1,1983
Ladyzhenskaja, O.A., The boundary value problems of Mathematical Physics, Applied Mathematical Sciences, Springer, New York, Vol. 49, 1985
Ling, L., and E.J. Kansa, A least-squares preconditioner for radial basis functions collocation methods, Adv. Comput. Math. 28 (2005), 31–54
Madych, W.R., and S.A. Nelson, Polyharmonic cardinal splines, J. Approx. Theory 60 (1990),141–156
Micchelli, C., C. Rabut, and F. Utreras, Using the refinement equation for the construction of pre-wavelets, III: Elliptic splines, Numer. Algorithms 1 (1991), 331–352
Rabut, C., Elementary polyharmonic cardinal B-splines, Numer. Algorithms 2 (1992), 39–46
Schaback, R., and H. Wendland, Kernel techniques: from machine learning to meshless methods, Acta Numerica (2006), 1–97
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer Science + Business Media B.V
About this chapter
Cite this chapter
Bacchelli, B., Bozzini, M. (2009). Particular Solution of Poisson Problems Using Cardinal Lagrangian Polyharmonic Splines. In: Ferreira, A.J.M., Kansa, E.J., Fasshauer, G.E., Leitão, V.M.A. (eds) Progress on Meshless Methods. Computational Methods in Applied Sciences, vol 11. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8821-6_1
Download citation
DOI: https://doi.org/10.1007/978-1-4020-8821-6_1
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-8820-9
Online ISBN: 978-1-4020-8821-6
eBook Packages: EngineeringEngineering (R0)