H2-convergence of least-squares kernel collocation methods

Ka Chun Cheung, Leevan Ling*, Robert Schaback

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

22 Citations (Scopus)
22 Downloads (Pure)


The strong-form asymmetric kernel-based collocation method, commonly referred to as the Kansa method, is easy to implement and hence is widely used for solving engineering problems and partial differential equations despite the lack of theoretical support. The simple least-squares (LS) formulation, on the other hand, makes the study of its solvability and convergence rather nontrivial. In this paper, we focus on general second order linear elliptic differential equations in Ω ⊂ Rd under Dirichlet boundary conditions. With kernels that reproduce Hm(Ω) and some smoothness assumptions on the solution, we provide conditions for a constrained LS method and a class of weighted LS algorithms to be convergent. Theoretically, for max(2, (d + 1)/2) ≤ ν ≤ m, we identify some Hν(Ω) convergent LS formulations that have an optimal error behavior like hm−ν For d ≤ 3, the proposed methods are optimal in H2(Ω). We demonstrate the effects of various . collocation settings on the respective convergence rates.

Original languageEnglish
Pages (from-to)614-633
Number of pages20
JournalSIAM Journal on Numerical Analysis
Issue number1
Publication statusPublished - 20 Feb 2018

Scopus Subject Areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Kansa method
  • Meshfree method
  • Overdetermined collocation
  • Radial basis function


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