### Abstract

We introduce a finite element construction for use on the class of convex, planar polygons and show that it obtains a quadratic error convergence estimate. On a convex n-gon, our construction produces 2n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, by transforming and combining a set of n(n + 1)/2 basis functions known to obtain quadratic convergence. This technique broadens the scope of the so-called 'serendipity' elements, previously studied only for quadrilateral and regular hexahedral meshes, by employing the theory of generalized barycentric coordinates. Uniform a priori error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges. Numerical evidence is provided on a trapezoidal quadrilateral mesh, previously not amenable to serendipity constructions, and applications to adaptive meshing are discussed.

Original language | English (US) |
---|---|

Pages (from-to) | 2691-2716 |

Number of pages | 26 |

Journal | Mathematics of Computation |

Volume | 83 |

Issue number | 290 |

State | Published - 2014 |

### Fingerprint

### Keywords

- Barycentric coordinates
- Finite element
- Serendipity

### ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics
- Computational Mathematics

### Cite this

*Mathematics of Computation*,

*83*(290), 2691-2716.

**Quadratic serendipity finite elements on polygons using generalized barycentric coordinates.** / Rand, Alexander; Gillette, Andrew; Bajaj, Chandrajit.

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 83, no. 290, pp. 2691-2716.

}

TY - JOUR

T1 - Quadratic serendipity finite elements on polygons using generalized barycentric coordinates

AU - Rand, Alexander

AU - Gillette, Andrew

AU - Bajaj, Chandrajit

PY - 2014

Y1 - 2014

N2 - We introduce a finite element construction for use on the class of convex, planar polygons and show that it obtains a quadratic error convergence estimate. On a convex n-gon, our construction produces 2n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, by transforming and combining a set of n(n + 1)/2 basis functions known to obtain quadratic convergence. This technique broadens the scope of the so-called 'serendipity' elements, previously studied only for quadrilateral and regular hexahedral meshes, by employing the theory of generalized barycentric coordinates. Uniform a priori error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges. Numerical evidence is provided on a trapezoidal quadrilateral mesh, previously not amenable to serendipity constructions, and applications to adaptive meshing are discussed.

AB - We introduce a finite element construction for use on the class of convex, planar polygons and show that it obtains a quadratic error convergence estimate. On a convex n-gon, our construction produces 2n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, by transforming and combining a set of n(n + 1)/2 basis functions known to obtain quadratic convergence. This technique broadens the scope of the so-called 'serendipity' elements, previously studied only for quadrilateral and regular hexahedral meshes, by employing the theory of generalized barycentric coordinates. Uniform a priori error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges. Numerical evidence is provided on a trapezoidal quadrilateral mesh, previously not amenable to serendipity constructions, and applications to adaptive meshing are discussed.

KW - Barycentric coordinates

KW - Finite element

KW - Serendipity

UR - http://www.scopus.com/inward/record.url?scp=84896549734&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84896549734&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84896549734

VL - 83

SP - 2691

EP - 2716

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 290

ER -