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

Carlos Zuppa

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

Abstract

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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.

How to cite

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Zuppa, 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

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  11. R.J. Renka, Multivariate interpolation of large sets of scattered data. ACM Trans. Math. Software14 (1988) 139–148.  
  12. L.L. Schumaker, Fitting surfaces to scattered data, in Approximation Theory II, Academic Press, Inc., New York (1970).  
  13. D.D. Shepard, A Two Dimensional Interpolation Function for Irregularly Spaced Data. Proc. 23rd Nat. Conf. ACM (1968).  
  14. R. Verfúrth, A note on polynomial approximation in Sobolev spaces. ESAIM: M2AN33 (1999) 715–719.  
  15. C. Zuppa, Error estimates for modified local Shepard's formulaes. Appl. Numer. Math. (to appear).  
  16. C. Zuppa, Good quality point sets and error estimates for moving least square approximations. Appl. Numer. Math.47 (2003) 575–585.  

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