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QUADRATIC SERENDIPITY FINITE ELEMENTS ON POLYGONS USING GENERALIZED BARYCENTRIC COORDINATES.

Alexander Rand1, Andrew Gillette2, Chandrajit Bajaj3

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Mathematics of Computation
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Summary
This summary is machine-generated.

This study presents a new finite element method for convex polygons, achieving quadratic convergence. The technique expands serendipity elements to more complex shapes, enhancing mesh adaptivity.

Keywords:
barycentric coordinatesfinite elementserendipity

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Area of Science:

  • Computational mathematics
  • Numerical analysis
  • Finite element methods

Background:

  • Existing finite element methods are limited in their application to complex polygonal meshes.
  • Serendipity elements, known for quadratic convergence, were previously restricted to quadrilaterals and hexahedra.

Purpose of the Study:

  • To develop a novel finite element construction for convex planar polygons.
  • To achieve quadratic error convergence estimates for these polygons.
  • To extend the applicability of serendipity elements to a broader class of meshes.

Main Methods:

  • A finite element construction is introduced for convex planar polygons.
  • It transforms and combines existing basis functions to create 2n basis functions for an n-gon.
  • The theory of generalized barycentric coordinates is employed to broaden serendipity element scope.

Main Results:

  • The construction yields a quadratic error convergence estimate.
  • Uniform a priori error estimates are established for convex quadrilaterals and other regular polygons.
  • Numerical evidence is provided on a trapezoidal quadrilateral mesh.

Conclusions:

  • The proposed finite element construction successfully applies to convex polygons, achieving quadratic convergence.
  • This method expands the utility of serendipity elements beyond traditional quadrilateral and hexahedral meshes.
  • The findings support applications in adaptive meshing for complex geometries.