Difference operators from interpolating moving least squares and their deviation from optimality
ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 5, pp. 883-908.

We consider the classical Interpolating Moving Least Squares (IMLS) interpolant as defined by Lancaster and Šalkauskas [Math. Comp. 37 (1981) 141-158] and compute the first and second derivative of this interpolant at the nodes of a given grid with the help of a basic lemma on Shepard interpolants. We compare the difference formulae with those defining optimal finite difference methods and discuss their deviation from optimality.

DOI : 10.1051/m2an:2005039
Classification : 39A70, 39A12, 65D05, 65D25
Mots clés : difference operators, moving least squares interpolation, order of approximation
@article{M2AN_2005__39_5_883_0,
     author = {Sonar, Thomas},
     title = {Difference operators from interpolating moving least squares and their deviation from optimality},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {883--908},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {5},
     year = {2005},
     doi = {10.1051/m2an:2005039},
     mrnumber = {2178566},
     zbl = {1085.39018},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2005039/}
}
TY  - JOUR
AU  - Sonar, Thomas
TI  - Difference operators from interpolating moving least squares and their deviation from optimality
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2005
SP  - 883
EP  - 908
VL  - 39
IS  - 5
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2005039/
DO  - 10.1051/m2an:2005039
LA  - en
ID  - M2AN_2005__39_5_883_0
ER  - 
%0 Journal Article
%A Sonar, Thomas
%T Difference operators from interpolating moving least squares and their deviation from optimality
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2005
%P 883-908
%V 39
%N 5
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2005039/
%R 10.1051/m2an:2005039
%G en
%F M2AN_2005__39_5_883_0
Sonar, Thomas. Difference operators from interpolating moving least squares and their deviation from optimality. ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 5, pp. 883-908. doi : 10.1051/m2an:2005039. http://www.numdam.org/articles/10.1051/m2an:2005039/

[1] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming and P. Krysl, Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Engrg. 139 (1996) 3-47. | Zbl

[2] J.P. Boyd, Chebyshev and Fourier Spectral Methods. Springer Verlag (1989). | Zbl

[3] B. Fornberg, Generation of Finite Difference Formulas on Arbitrarily Spaced Grids. Math. Comp. 51 (1988) 699-706. | Zbl

[4] B. Fornberg, A Practical Guide to Pseudospectral Methods. Cambridge University Press (1996). | MR | Zbl

[5] J. Fürst and Th. Sonar, On meshless collocation approximations of conservation laws: preliminary investigations on positive schemes and dissipation models. ZAMM Z. Angew. Math. Mech. 81 (2001) 403-415. | Zbl

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

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

[8] P. Lancaster and K. Šalkauskas, Curve and Surface Fitting: An Introduction. Academic Press (1986). | MR | Zbl

[9] T. Liszka and J. Orkisz, The finite difference method at arbitrary irregular grids and its application in applied mechanics. Comput. Structures 11 (1980) 83-95. | Zbl

[10] H. Netuzylov, Th. Sonar and W. Yomsatieankul, Finite difference operators from moving least squares interpolation. Manuscript, Institut Computational Mathematics, TU Braunschweig (2004).

[11] N. Perrone and R. Kao, A general finite difference method for arbitrary meshes. Comput. Structures 5 (1975) 45-58.

[12] W. Schönauer, Generation of difference and error formulae of arbitrary consistency order on an unstructured grid. ZAMM Z. Angew. Math. Mech. 78 (1998) S1061-S1062. | Zbl

[13] L. Theilemann, Ein gitterfreies differenzenverfahren. Doktorarbeit, Institut für Aerodynamik und Gasdynamik, Universität Stuttgart (1983).

[14] W. Yomsatieankul, Th. Sonar and H. Netuzhylov, Spatial difference operators from moving least squares interpolation. Manuscript, Institut Computational Mathematics, TU Braunschweig (2004).

Cité par Sources :