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J A Reeger1, B Fornberg2, M L Watts1

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This study extends an efficient surface integral method using local radial basis function interpolation. The generalization reduces computational complexity for quadrature on arbitrary smooth closed surfaces.

Keywords:
quadratureradial basis function-finite differencesradial basis functionssurface integral

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Area of Science:

  • Numerical Analysis
  • Computational Geometry

Background:

  • Numerical approximation of definite integrals (quadrature) typically uses polynomial interpolation.
  • Extending polynomial interpolation to higher dimensions is computationally expensive and unstable.
  • Existing methods efficiently compute surface integrals on spheres using local radial basis functions.

Purpose of the Study:

  • To generalize an efficient quadrature method for surface integrals.
  • To extend the application of local radial basis function interpolation to arbitrary smooth closed surfaces.

Main Methods:

  • Generalizing a previously established method for spherical surface integrals.
  • Employing local radial basis function interpolation for efficient quadrature weight generation.
  • Applying the method to arbitrary smooth closed surfaces.

Main Results:

  • The generalized method reduces computational complexity for surface integral approximation.
  • Local radial basis function interpolation proves effective on various smooth closed surfaces.
  • The approach provides a stable and efficient alternative to traditional methods.

Conclusions:

  • The generalization of local radial basis function interpolation offers an efficient approach to surface quadrature.
  • This method enhances the computation of integrals on complex smooth closed surfaces.
  • The study provides a valuable tool for numerical integration in higher dimensions.