Collocation and optimization initialization

E. J. Kansa*, Leevan LING

*Corresponding author for this work

Research output: Chapter in book/report/conference proceedingConference contributionpeer-review

1 Citation (Scopus)

Abstract

Integrated volumetric methods such as finite elements and their “meshless” variations are typically smoother than the strong form finite difference and radial basis function collocation methods. Numerical methods decrease their convergence rates with successively higher orders of differentiation along with improved conditioning. In contrast, increasing order of integration increases the convergence rate at the expense of poorer conditioning. In the study presented, a two-dimensional Poisson equation with exponential dependency is solved. The solution of the point collocation problem becomes the initial estimate for an integrated volumetric minimization process. Global, rather than local integration, is used since there is no need to construct any meshes for integration as done in the “meshless” finite element analogs. The root mean square (RMS) errors are compared. By pushing the shape parameter to very large values, using extended precision, the RMS errors show that spatial refinement benefits are relatively small compared to pushing shape parameters to increasing larger values. The improved Greedy Algorithm was used to optimize the set of data and evaluation centers for various shape parameters. Finally, extended arithmetic precision is used to push the range of the shape parameters.

Original languageEnglish
Title of host publicationWIT Transactions on Modelling and Simulation
EditorsA.H-D. Cheng, Carlos A. Brebbia
PublisherWITPress
Pages55-62
Number of pages8
ISBN (Electronic)9781845648961
DOIs
Publication statusPublished - 8 Sep 2014
Event37th Boundary Element Methods Conference, BEM 2014 - New Forest, United Kingdom
Duration: 8 Sep 201410 Sep 2014

Publication series

NameWIT Transactions on Modelling and Simulation
Volume57
ISSN (Print)1743-355X

Conference

Conference37th Boundary Element Methods Conference, BEM 2014
Country/TerritoryUnited Kingdom
CityNew Forest
Period8/09/1410/09/14

Scopus Subject Areas

  • Modelling and Simulation
  • Computational Mathematics

User-Defined Keywords

  • Global minimization
  • Meshless radial basis functions
  • Multiquadric
  • Partial differential equations
  • Strong and weak formulation

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