# 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

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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

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- C. Zuppa, Good quality point sets and error estimates for moving least square approximations. Appl. Numer. Math.47 (2003) 575–585. Zbl1040.65034

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