Finite difference operators from moving least squares interpolation
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 5, p. 959-974

In a foregoing paper [Sonar, ESAIM: M2AN 39 (2005) 883-908] we analyzed the Interpolating Moving Least Squares (IMLS) method due to Lancaster and Šalkauskas with respect to its approximation powers and derived finite difference expressions for the derivatives. In this sequel we follow a completely different approach to the IMLS method given by Kunle [Dissertation (2001)]. As a typical problem with IMLS method we address the question of getting admissible results at the boundary by introducing “ghost points”. Most interesting in IMLS methods are the finite difference operators which can be computed from different choices of basis functions and weight functions. We present a way of deriving these discrete operators in the spatially one-dimensional case. Multidimensional operators can be constructed by simply extending our approach to higher dimensions. Numerical results ranging from 1-d interpolation to the solution of PDEs are given.

DOI : https://doi.org/10.1051/m2an:2007042
Classification:  65M06,  65M60,  65F05
Keywords: difference operators, moving least squares interpolation, order of approximation
@article{M2AN_2007__41_5_959_0,
     author = {Netuzhylov, Hennadiy and Sonar, Thomas and Yomsatieankul, Warisa},
     title = {Finite difference operators from moving least squares interpolation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {5},
     year = {2007},
     pages = {959-974},
     doi = {10.1051/m2an:2007042},
     zbl = {pre05289357},
     mrnumber = {2363891},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2007__41_5_959_0}
}
Netuzhylov, Hennadiy; Sonar, Thomas; Yomsatieankul, Warisa. Finite difference operators from moving least squares interpolation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 5, pp. 959-974. doi : 10.1051/m2an:2007042. http://www.numdam.org/item/M2AN_2007__41_5_959_0/

[1] G.H. Golub and C.F. Van Loan, Matrix Computations. Johns Hopkins Univ. Press (1996). | MR 1417720 | Zbl 0865.65009

[2] M. Kunle, Entwicklung und Untersuchung von Moving Least Square Verfahren zur numerischen Simulation hydrodynamischer Gleichungen. Dissertation, Fakultät für Physik, Eberhard-Karls-Universität zu Tübingen (2001).

[3] P. Lancaster and K. Šalkauskas, Surfaces generated by moving least square methods. Math. Comp. 37 (1981) 141-158. | Zbl 0469.41005

[4] P. Lancaster and K. Šalkauskas, Curve and Surface Fitting - An Introduction. Academic Press (1986). | MR 1001969 | Zbl 0649.65012

[5] H. Netuzhylov, Meshfree collocation solution of Boundary Value Problems via Interpolating Moving Least Squares. Comm. Num. Meth. Engng. 22 (2006) 893-899. | Zbl 1105.65356

[6] O. Nowak and T. Sonar, Upwind and ENO strategies in Interpolating Moving Least Squares methods (in preparation).

[7] T. Sonar, Difference operators from interpolating moving least squares and their deviation from optimality. ESAIM: M2AN 39 (2005) 883-908. | Numdam | Zbl 1085.39018