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Convergence analysis of a hierarchical enrichment of Dirichlet boundary conditions in a mesh‐free method
Author(s) -
Han Weimin,
Wagner Gregory J.,
Liu Wing Kam
Publication year - 2001
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.336
Subject(s) - convergence (economics) , interpolation (computer graphics) , mathematics , boundary (topology) , dirichlet distribution , polygon mesh , boundary value problem , simple (philosophy) , mathematical optimization , computer science , algorithm , mathematical analysis , geometry , artificial intelligence , philosophy , epistemology , economics , economic growth , motion (physics)
Implementation of Dirichlet boundary conditions in mesh‐free methods is problematic. In Wagner and Liu ( International Journal for Numerical Methods in Engineering 2001; 50 :507), a hierarchical enrichment technique is introduced that allows a simple implementation of the Dirichlet boundary conditions. In this paper, we provide some error analysis for the hierarchical enrichment mesh‐free technique. We derive optimal order error estimates for the hierarchical enrichment mesh‐free interpolants. For one‐dimensional elliptic boundary value problems, we can directly apply the interpolation error estimates to obtain error estimates for the mesh‐free solutions. For higher‐dimensional problems, derivation of error estimates for the mesh‐free solutions depends on the availability of an inverse inequality. Numerical examples in 1D and 2D are included showing the convergence behaviour of mesh‐free interpolants and mesh‐free solutions when the hierarchical enrichment mesh‐free technique is employed. Copyright © 2001 John Wiley & Sons, Ltd.