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Related Experiment Videos

Multivariate approximation by locally blended univariate interpolants.

G Birkhoff1, J C Cavendish, W J Gordon

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

Proceedings of the National Academy of Sciences of the United States of America
|September 1, 1974
PubMed
Summary

Researchers developed new finite elements for approximating smooth functions in rectangular domains. These elements are suitable for rectangular cell decompositions and have analogous 3D counterparts.

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Partial Differential Equations

Background:

  • Approximating smooth functions is crucial in numerical simulations.
  • Existing finite elements may not be optimal for rectangular geometries.
  • Efficient methods are needed for complex domain discretizations.

Purpose of the Study:

  • To introduce novel finite elements specifically designed for rectangular polygons.
  • To demonstrate the suitability of these elements for approximating smooth functions.
  • To extend the concept to three-dimensional rectangular cells (bricks).

Main Methods:

  • Development of new finite element formulations for 2D rectangular cells.
  • Derivation and analysis of key properties of these elements.

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  • Construction of analogous 3D finite elements (bricks).

    • Main Results:

    • The proposed finite elements are simple and well-suited for smooth function approximation.
    • Key properties supporting their effectiveness were mathematically derived.
    • Successful construction of analogous three-dimensional elements was achieved.

    Conclusions:

    • The new finite elements offer a promising approach for simulations involving rectangular domains.
    • The methodology provides a foundation for developing specialized elements in computational mathematics.
    • The 3D extension indicates broader applicability in numerical methods.