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Tricubic polynomial interpolation.

G Birkhoff1

  • 1Mathematics Department, Harvard University, Cambridge, Mass. 02138.

Proceedings of the National Academy of Sciences of the United States of America
|June 1, 1971
PubMed
Summary

Researchers developed a novel finite element for accurate function approximation. This tricubic element ensures continuous differentiability across triangulated domains, enhancing interpolation accuracy for smooth functions.

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Finite Element Methods

Background:

  • Accurate approximation of smooth functions is crucial in various scientific and engineering fields.
  • Existing finite elements may lack sufficient continuity or accuracy for certain applications.
  • Triangulated domains are common in computational modeling.

Purpose of the Study:

  • To introduce a new triangular finite element with enhanced approximation properties.
  • To develop an interpolation scheme utilizing this novel element for smooth function approximation.
  • To achieve continuous differentiability across triangulated domains.

Main Methods:

  • Description of a 12-parameter family of quartic polynomial functions termed "tricubic" elements.
  • The tricubic variation is cubic along lines parallel to the triangle's sides.
  • Development of an interpolation scheme based on these tricubic elements.

Main Results:

  • The proposed tricubic finite element accurately approximates smooth functions.
  • The interpolation scheme ensures continuous differentiability on each triangular element.
  • High accuracy is achieved for functions defined on triangulated domains.

Conclusions:

  • The novel tricubic finite element offers a powerful tool for accurate function approximation.
  • This method provides a continuously differentiable interpolant over triangulated domains.
  • The approach enhances the fidelity of numerical simulations relying on function interpolation.

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