Hermite and Bernstein Style Basis Functions for Cubic Serendipity Spaces on Squares and Cubes

TL;DR

This paper introduces new Hermite and Bernstein basis functions for cubic serendipity finite element spaces on squares and cubes, reducing the number of basis functions while maintaining error estimates, and facilitating integration into existing finite element and isogeometric analysis frameworks.

Contribution

The authors develop new geometric decompositions of cubic serendipity spaces using Hermite and Bernstein bases, linking to finite element degrees of freedom and domain geometry, with compatibility for existing codes.

Findings

Basis functions have a canonical relationship to degrees of freedom.

Reduced basis size (12 for squares, 32 for cubes) compared to tensor product spaces.

Compatible with isogeometric analysis and existing finite element implementations.

Abstract

We introduce new Hermite-style and Bernstein-style geometric decompositions of the cubic order serendipity finite element spaces $S_3(I^2)$ and $S_3(I^3)$, as defined in the recent work of Arnold and Awanou [Found. Comput. Math. 11 (2011), 337--344]. The serendipity spaces are substantially smaller in dimension than the more commonly used bicubic and tricubic Hermite tensor product spaces - 12 instead of 16 for the square and 32 instead of 64 for the cube - yet are still guaranteed to obtain cubic order \textit{a priori} error estimates in $H^1$ norm when used in finite element methods. The basis functions we define have a canonical relationship both to the finite element degrees of freedom as well as to the geometry of their graphs; this means the bases may be suitable for applications employing isogeometric analysis where domain geometry and functions supported on the domain are described by the same basis functions. Moreover, the basis functions are linear combinations of the commonly used bicubic and tricubic polynomial Bernstein or Hermite basis functions, allowing their rapid incorporation into existing finite element codes.