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.
Mots-clés : error estimates, Shepard's formulas, Jackson inequalities, Sobolev spaces
@article{M2AN_2003__37_6_973_0,
author = {Zuppa, Carlos},
title = {Error estimates for modified local {Shepard's} formulas in {Sobolev} spaces},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {973--989},
publisher = {EDP-Sciences},
volume = {37},
number = {6},
year = {2003},
doi = {10.1051/m2an:2003063},
zbl = {1074.65125},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2003063/}
}
TY - JOUR AU - Zuppa, Carlos TI - Error estimates for modified local Shepard's formulas in Sobolev spaces JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2003 SP - 973 EP - 989 VL - 37 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2003063/ DO - 10.1051/m2an:2003063 LA - en ID - M2AN_2003__37_6_973_0 ER -
%0 Journal Article %A Zuppa, Carlos %T Error estimates for modified local Shepard's formulas in Sobolev spaces %J ESAIM: Modélisation mathématique et analyse numérique %D 2003 %P 973-989 %V 37 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2003063/ %R 10.1051/m2an:2003063 %G en %F M2AN_2003__37_6_973_0
Zuppa, Carlos. Error estimates for modified local Shepard's formulas in Sobolev spaces. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 6, pp. 973-989. doi : 10.1051/m2an:2003063. http://www.numdam.org/articles/10.1051/m2an:2003063/
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