# Error estimates for Modified Local Shepard's Formulas in Sobolev spaces

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 37, Issue: 6, page 973-989
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topZuppa, Carlos. "Error estimates for Modified Local Shepard's Formulas in Sobolev spaces." ESAIM: Mathematical Modelling and Numerical Analysis 37.6 (2010): 973-989. <http://eudml.org/doc/194200>.

@article{Zuppa2010,

abstract = {
Interest in meshfree methods in solving boundary-value problems has grown
rapidly in recent years. A meshless method that has attracted considerable
interest in the community of computational mechanics is built around the
idea of modified local Shepard's partition of unity. For these kinds of
applications it is fundamental to analyze the order of the approximation in
the context of Sobolev spaces. In this paper, we study two different
techniques for building modified local Shepard's formulas, and we provide a
theoretical analysis for error estimates of the approximation in Sobolev
norms. We derive Jackson-type inequalities for h-p cloud functions
using the first construction. These estimates are important in the analysis
of Galerkin approximations based on local Shepard's formulas or h-p
cloud functions.
},

author = {Zuppa, Carlos},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Error estimates; Shepard's formulas; Jackson
inequalities; Sobolev spaces.; Jackson inequalities; Sobolev spaces; Galerkin method; finite element method; meshfree methods},

language = {eng},

month = {3},

number = {6},

pages = {973-989},

publisher = {EDP Sciences},

title = {Error estimates for Modified Local Shepard's Formulas in Sobolev spaces},

url = {http://eudml.org/doc/194200},

volume = {37},

year = {2010},

}

TY - JOUR

AU - Zuppa, Carlos

TI - Error estimates for Modified Local Shepard's Formulas in Sobolev spaces

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 37

IS - 6

SP - 973

EP - 989

AB -
Interest in meshfree methods in solving boundary-value problems has grown
rapidly in recent years. A meshless method that has attracted considerable
interest in the community of computational mechanics is built around the
idea of modified local Shepard's partition of unity. For these kinds of
applications it is fundamental to analyze the order of the approximation in
the context of Sobolev spaces. In this paper, we study two different
techniques for building modified local Shepard's formulas, and we provide a
theoretical analysis for error estimates of the approximation in Sobolev
norms. We derive Jackson-type inequalities for h-p cloud functions
using the first construction. These estimates are important in the analysis
of Galerkin approximations based on local Shepard's formulas or h-p
cloud functions.

LA - eng

KW - Error estimates; Shepard's formulas; Jackson
inequalities; Sobolev spaces.; Jackson inequalities; Sobolev spaces; Galerkin method; finite element method; meshfree methods

UR - http://eudml.org/doc/194200

ER -

## References

top- R.A. Adams, Sobolev Spaces. Academic Press, Inc., Orlando (1975).
- S.C. Brener and L.R. Scott, The Mathematical Theory of Finite Elements Methods. Springer-Verlag, New York (1994).
- P.G. Ciarlet, The Finite Elements Method for Elliptic Problems. North-Holland, Amsterdam (1978). Zbl0383.65058
- C.A. Duarte and J.T. Oden, Hp clouds-a meshless method to solve boundary-value problems. Technical Report 95-05, TICAM, The University of Texas at Austin (1995).
- C.A. Duarte and J.T. Oden, H-p clouds-an h-p meshless method. Numer. Methods Partial Differential Equations1 (1996) 1–34. Zbl0869.65069
- C.A.M. Duarte, T.J. Liszka and W.W. Tworzydlo, hp-meshless cloud method. Comput. Methods Appl. Mech. Engrg.139 (1996) 263–288. Zbl0893.73077
- R.G. Durán, On polynomial approximation in Sobolev spaces. SIAM J. Numer. Anal.20 (1983) 985–988.
- W. Han and X. Meng, Error analysis of the reproducing kernel particle method. Comput. Methods Appl. Mech. Engrg.190 (2001) 6157–6181. Zbl0992.65119
- Y.Y. Lu, T. Belyschko and L. Gu, Element-free Galerkin methods. Internat. J. Numer. Methods Engrg.37 (1994) 229–256. Zbl0796.73077
- E. Oñate, R. Taylor, O.C. Zienkiewicz and S. Idelshon, Moving least square approximations for the solutions of differential equations. Technical Report, CIMNE, Santa Fé, Argentina (1995).
- R.J. Renka, Multivariate interpolation of large sets of scattered data. ACM Trans. Math. Software14 (1988) 139–148. Zbl0642.65006
- L.L. Schumaker, Fitting surfaces to scattered data, in Approximation Theory II, Academic Press, Inc., New York (1970).
- D.D. Shepard, A Two Dimensional Interpolation Function for Irregularly Spaced Data. Proc. 23rd Nat. Conf. ACM (1968).
- R. Verfúrth, A note on polynomial approximation in Sobolev spaces. ESAIM: M2AN33 (1999) 715–719.
- C. Zuppa, Error estimates for modified local Shepard's formulaes. Appl. Numer. Math. (to appear). Zbl1074.65125
- C. Zuppa, Good quality point sets and error estimates for moving least square approximations. Appl. Numer. Math.47 (2003) 575–585. Zbl1040.65034

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.