Arbitrary placement of secondary nodes, and error control, in the meshless local Petrov-Galerkin (MLPG) method

H. G. Kim, S. N. Atluri

Research output: Contribution to journalArticlepeer-review

62 Scopus citations


The truly meshless local Petrov-Galerkin (MLPG) method holds a great promise in solving boundary value problems, using a local symmetric weak form as a natural approach. In the present paper, in the context of MLPG and the meshless interpolation of a moving least squares (MLS) type, a method which uses primary and secondary nodes in the domain and on the global boundary is introduced, in order to improve the accuracy of solution. The secondary nodes can be placed at any location where one needs to obtain a better resolution. The sub-domains for the shape functions in the MLS approximation are defined only from the primary nodes, and the secondary nodes use the same sub-domains. The shape functions based on the MLS approximation, in an integration domain, have a single type of a rational function, which reduces the difficulty of numerical integration to evaluate the weak form. The present method is very useful in an adaptive calculation, because the secondary nodes can be easily added and/or moved without an additional mesh. The essential boundary conditions can be imposed exactly, and non-convex boundaries can be treated without special techniques. Several numerical examples are presented to illustrate the performance of the present method.

Original languageEnglish
Pages (from-to)10-30
Number of pages21
JournalCMES - Computer Modeling in Engineering and Sciences
Issue number3
StatePublished - 2000


  • Local symmetric weak form
  • MLPG method
  • MLS
  • Meshless method
  • Primary node
  • Secondary node


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