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shepard's formulas

ESAIM: Mathematical Modelling and Numerical Analysis (1)

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Error estimates for Modified Local Shepard's Formulasin Sobolev spaces
Carlos Zuppa
Journal:

ESAIM: Mathematical Modelling and Numerical Analysis / Volume 37 / Issue 6 / November 2003

Published online by Cambridge University Press:

15 November 2003, pp. 973-989

Print publication:

November 2003
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Interest in meshfree methods in solving boundary-value problems has grownrapidly in recent years. A meshless method that has attracted considerableinterest in the community of computational mechanics is built around theidea of modified local Shepard's partition of unity. For these kinds ofapplications it is fundamental to analyze the order of the approximation inthe context of Sobolev spaces. In this paper, we study two differenttechniques for building modified local Shepard's formulas, and we provide atheoretical analysis for error estimates of the approximation in Sobolevnorms. We derive Jackson-type inequalities for h-p cloud functionsusing the first construction. These estimates are important in the analysisof Galerkin approximations based on local Shepard's formulas or h-pcloud functions.