Reproducing kernel hierarchical partition of unity, part I - Formulation and theory

Shaofan Li, Wing Kam Liu*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

156 Scopus citations


This work is concerned with developing the hierarchical basis for meshless methods. A reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart. To form such hierarchical partition, a class of basic wavelet functions are introduced. Based upon the built-in consistency conditions, the differential consistency conditions for the hierarchical kernel functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. For a special interpolant with different combinations of the hierarchical kernels, a synchronized convergence effect may be observed. Being different from the conventional Legendre function based p-type hierarchical basis, the new hierarchical basis is an intrinsic pseudo-spectral basis, which can remain as a partition of unity in a local region, because the discrete wavelet kernels form a 'partition of nullity'. These newly developed kernels can be used as the multiscale basis to solve partial differential equations in numerical computation as a p-type refinement.

Original languageEnglish (US)
Pages (from-to)251-288
Number of pages38
JournalInternational Journal for Numerical Methods in Engineering
Issue number3
StatePublished - May 30 1999


  • Meshless hierarchical partition of unity
  • Moving least-squares interpolant
  • Reproducing kernel particle method
  • Synchronized convergence
  • Wavelet methods

ASJC Scopus subject areas

  • Numerical Analysis
  • Engineering(all)
  • Applied Mathematics


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