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Error estimates for modified local Shepard’s formulas in Sobolev spaces. (English) Zbl 1074.65125

Summary: 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 the modified local partition of unity of D. D. Shepard [A two-dimensional interpolation function for irregularly spaced data. Proc. 23rd Nat. Conf. ACM (1968)]. For these kinds of applications it is fundamental to analyze the order of the approximation in the context of Sobolev spaces.
We study two different techniques for building the 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.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations

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References:

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