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The serendipity family of finite elements
Douglas N. Arnold, Gerard Awanou
TL;DR
Serendipity finite elements lacked a simple, systematic definition across dimensions, especially for higher-degree three-dimensional elements. The paper defines their shape-function spaces and face-based degrees of freedom dimension-independently, then proves unisolvence and gives a geometric decomposition. The result is a unified finite element family with degrees of freedom that determine face restrictions and support continuity after assembly.
Problem
Serendipity elements, particularly in three dimensions and higher degrees, lacked a simple systematic definition of their shape functions and degrees of freedom.
Method
The paper defines serendipity polynomial spaces and face-associated moment degrees of freedom for general dimension n and degree r.
Results
The specified degrees of freedom are unisolvent for Sr(In), and their face and subface data determine restrictions needed for assembled continuity.
Takeaways & Limitations
The construction provides a dimension-independent serendipity family together with a geometric decomposition into face-associated subspaces.
Abstract
from arXiv · showhide
We give a new, simple, dimension-independent definition of the serendipity finite element family. The shape functions are the span of all monomials which are linear in at least s-r of the variables where s is the degree of the monomial or, equivalently, whose superlinear degree (total degree with respect to variables entering at least quadratically) is at most r. The degrees of freedom are given by moments of degree at most r-2d on each face of dimension d. We establish unisolvence and a geometric decomposition of the space.
1. Introduction
Serendipity elements reduce the dimension of tensor-product Lagrange elements while retaining C0 continuity, but their higher-dimensional structure has been difficult to describe systematically. The paper introduces a simple, dimension-independent definition of their shape functions and degrees of freedom, and proves the resulting count and unisolvence.
- Motivation: Serendipity elements provide lower-dimensional C0-conforming finite element subspaces than tensor-product Lagrange elements on parallelepiped meshes.They are especially established in two dimensions and used to a lesser extent in three dimensions.
- Degrees of freedom: For tensor-product elements, face degrees of freedom use polynomial moments while vertices use evaluations, and their total number equals dim Qr(In).The count follows from the number of d-dimensional faces and a binomial expansion.
- Serendipity construction: The serendipity construction preserves boundary degrees of freedom while removing interior degrees of freedom and choosing a smaller shape-function space.This produces a lower-dimensional space intended to retain C0 continuity without much loss of accuracy.
- Motivation: Three-dimensional serendipity elements are commonly discussed only at low orders, while the higher-degree pattern and shape-function space are not evident.The cited low-order examples include the 20-node and possibly 32-node bricks, with no interior degrees of freedom for r = 2, 3.
- Contribution: The paper defines the serendipity family for general dimension n and degree r through a polynomial space and degrees of freedom associated with every cube face.The definition is presented as self-contained and dimension-independent.
2. Shape functions and degrees of freedom
The paper defines serendipity shape functions in any dimension using monomial linearity or, equivalently, bounded superlinear degree, and assigns face moments that are proven unisolvent. This framework recovers familiar low-dimensional spaces while specifying their higher-dimensional structure.
- Shape functions: The space S_r(I_n) is spanned by monomials of degree s that are linear in at least s-r variables.Equivalently, it consists of polynomials whose superlinear degree is at most r.
- Shape functions: The two definitions are equivalent because a monomial linear in l variables satisfies deg p = deg2 p + l.The paper notes that neither formulation appears to have been previously given in the literature.
- Shape functions: In two dimensions, the construction recovers the usual serendipity shape functions, while in three dimensions it adds selected monomials of degrees r+1 and r+2 to P_r(I_3).The added degree-r+1 monomials are linear in at least one variable; the degree-r+2 monomials are linear in at least two.
- Degrees of freedom: The proposed degrees of freedom are unisolvent for S_r(I_n), with their number equal to the dimension of the space.The proof uses induction on dimension: vanishing face data implies vanishing restrictions, then vanishing on all faces, and finally vanishing in the interior.
- Degrees of freedom: Face and subface degrees of freedom determine each face restriction, ensuring continuity when serendipity finite elements are assembled.This provides the C0 continuity property of the assembled finite element function.
3. Geometric decomposition
The serendipity space is decomposed directly into subspaces associated with the cube’s faces. Each face subspace is formed by multiplying a face-supported bubble function by polynomials in the face variables, and the decomposition supports explicit local bases.
- The cube’s faces are organized by dimension, with each face determined by fixing n−d coordinates to ±1.
- A face bubble function vanishes on cube faces that do not contain the selected face and is strictly positive on its relative interior.
- For a d-dimensional face, the associated subspace consists of its bubble function multiplied by polynomials of degree at most r−2d in the d unfixed variables.
- The face-associated subspaces form a direct-sum decomposition of the serendipity space.
- The proof expands monomials by separating constant, linear, and superlinear variables, then assigns resulting terms to suitable face subspaces.