Error estimates for modified local Shepard's formulas in Sobolev spaces
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 37 (2003) no. 6, pp. 973-989.

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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.

DOI : 10.1051/m2an:2003063
Classification : 41A10, 41A17, 65N15, 65N30
Mots-clés : error estimates, Shepard's formulas, Jackson inequalities, Sobolev spaces 
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     title = {Error estimates for modified local {Shepard's} formulas in {Sobolev} spaces},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {973--989},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {6},
     year = {2003},
     doi = {10.1051/m2an:2003063},
     zbl = {1074.65125},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2003063/}
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Zuppa, Carlos. Error estimates for modified local Shepard's formulas in Sobolev spaces. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 37 (2003) no. 6, pp. 973-989. doi : 10.1051/m2an:2003063. https://geodesic-test.mathdoc.fr/articles/10.1051/m2an:2003063/

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