A gradient reproducing kernel collocation method for boundary value problems

Sheng Wei Chi, Jiun Shyan Chen*, Hsin Yun Hu, P Yang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

38 Scopus citations

Abstract

The earlier work in the development of direct strong form collocation methods, such as the reproducing kernel collocation method (RKCM), addressed the domain integration issue in the Galerkin type meshfree method, such as the reproducing kernel particle method, but with increased computational complexity because of taking higher order derivatives of the approximation functions and the need for using a large number of collocation points for optimal convergence. In this work, we intend to address the computational complexity in RKCM while achieving optimal convergence by introducing a gradient reproduction kernel approximation. The proposed gradient RKCM reduces the order of differentiation to the first order for solving second-order PDEs with strong form collocation. We also show that, different from the typical strong form collocation method where a significantly large number of collocation points than the number of source points is needed for optimal convergence, the same number of collocation points and source points can be used in gradient RKCM. We also show that the same order of convergence rates in the primary unknown and its first-order derivative is achieved, owing to the imposition of gradient reproducing conditions. The numerical examples are given to verify the analytical prediction.

Original languageEnglish
Pages (from-to)1381-1402
Number of pages22
JournalInternational Journal for Numerical Methods in Engineering
Volume93
Issue number13
DOIs
StatePublished - 30 Mar 2013

Keywords

  • Gradient reproducing kernel approximation
  • Reproducing kernel collocation method
  • Strong form collocation
  • Weighted collocation method

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