# Difference operators from interpolating moving least squares and their deviation from optimality

- Volume: 39, Issue: 5, page 883-908
- ISSN: 0764-583X

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topSonar, Thomas. "Difference operators from interpolating moving least squares and their deviation from optimality." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.5 (2005): 883-908. <http://eudml.org/doc/245225>.

@article{Sonar2005,

abstract = {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.},

author = {Sonar, Thomas},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {difference operators; moving least squares interpolation; order of approximation; Difference operators; Shepard interpolant},

language = {eng},

number = {5},

pages = {883-908},

publisher = {EDP-Sciences},

title = {Difference operators from interpolating moving least squares and their deviation from optimality},

url = {http://eudml.org/doc/245225},

volume = {39},

year = {2005},

}

TY - JOUR

AU - Sonar, Thomas

TI - Difference operators from interpolating moving least squares and their deviation from optimality

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2005

PB - EDP-Sciences

VL - 39

IS - 5

SP - 883

EP - 908

AB - 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.

LA - eng

KW - difference operators; moving least squares interpolation; order of approximation; Difference operators; Shepard interpolant

UR - http://eudml.org/doc/245225

ER -

## References

top- [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. Zbl0891.73075
- [2] J.P. Boyd, Chebyshev and Fourier Spectral Methods. Springer Verlag (1989). Zbl0681.65079
- [3] B. Fornberg, Generation of Finite Difference Formulas on Arbitrarily Spaced Grids. Math. Comp. 51 (1988) 699–706. Zbl0701.65014
- [4] B. Fornberg, A Practical Guide to Pseudospectral Methods. Cambridge University Press (1996). Zbl0844.65084MR1386891
- [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. Zbl0985.65123
- [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. Zbl0469.41005
- [8] P. Lancaster and K. Šalkauskas, Curve and Surface Fitting: An Introduction. Academic Press (1986). Zbl0649.65012MR1001969
- [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. Zbl0427.73077
- [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. Zbl0925.65175
- [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).

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