Reproducing kernel element method. Part I: Theoretical formulation

Wing Kam Liu*, Weimin Han, Hongsheng Lu, Shaofan Li, Jian Cao

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

143 Scopus citations


In this paper and its sequels, we introduce and analyze a new class of methods, collectively called the reproducing kernel element method (RKEM). The central idea in the development of the new method is to combine the strengths of both finite element methods (FEM) and meshfree methods. Two distinguished features of RKEM are: the arbitrarily high order smoothness and the interpolation property of the shape functions. These properties are desirable especially in solving Galerkin weak forms of higher order partial differential equations and in treating Dirichlet boundary conditions. So unlike the FEM, there is no need for special treatment with the RKEM in solving high order equations. Compared to meshfree methods, Dirichlet boundary conditions do not present any difficulty in using the RKEM. A rigorous error analysis and convergence study of the method are presented. The performance of the method is illustrated and assessed through some numerical examples.

Original languageEnglish (US)
Pages (from-to)933-951
Number of pages19
JournalComputer Methods in Applied Mechanics and Engineering
Issue number12-14
StatePublished - Mar 26 2004


  • Approximation theory
  • Finite element method
  • Meshfree method
  • Reproducing kernel element method

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Physics and Astronomy(all)
  • Computer Science Applications


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