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A construction of canonical nonconforming finite element spaces for elliptic equations of any order in any dimension

Jia Li, Shuonan Wu

TL;DR

The paper solves the open problem of constructing canonical $H^m$-nonconforming finite elements on $n$-dimensional simplices for all $m,n\ge1$ by introducing a universal, layer-structured DOF system and a shape space $P_T^{(m,n)}$ augmented with nonconforming bubbles. An integral-type representation and single-variable composition bubbles enable a clean unisolvence proof by induction on $m$, while preserving consistency with MWX elements for $m\le n$ and reducing to conforming behavior when $n=1$. The resulting elements are applicable to $2m$-th order elliptic PDEs, with discontinuous Galerkin-like weak continuity and robust error estimates; numerical tests in 2D for $m=3,4$ validate linear convergence in the broken $H^m$ norm and expected rates in lower norms under smooth and singular data. This work significantly extends nonconforming element design to arbitrary $m$ and $n$, offering a compact, low-DOF alternative suitable for high-order elliptic problems and potentially enabling VEM extensions.

Abstract

A unified construction of canonical $H^m$-nonconforming finite elements is developed for $n$-dimensional simplices for any $m, n \geq 1$. Consistency with the Morley-Wang-Xu elements [Math. Comp. 82 (2013), pp. 25-43] is maintained when $m \leq n$. In the general case, the degrees of freedom and the shape function space exhibit well-matched multi-layer structures that ensure their alignment. Building on the concept of the nonconforming bubble function, the unisolvence is established using an equivalent integral-type representation of the shape function space and by applying induction on $m$. The corresponding nonconforming finite element method applies to $2m$-th order elliptic problems, with numerical results for $m=3$ and $m=4$ in 2D supporting the theoretical analysis.

A construction of canonical nonconforming finite element spaces for elliptic equations of any order in any dimension

TL;DR

The paper solves the open problem of constructing canonical -nonconforming finite elements on -dimensional simplices for all by introducing a universal, layer-structured DOF system and a shape space augmented with nonconforming bubbles. An integral-type representation and single-variable composition bubbles enable a clean unisolvence proof by induction on , while preserving consistency with MWX elements for and reducing to conforming behavior when . The resulting elements are applicable to -th order elliptic PDEs, with discontinuous Galerkin-like weak continuity and robust error estimates; numerical tests in 2D for validate linear convergence in the broken norm and expected rates in lower norms under smooth and singular data. This work significantly extends nonconforming element design to arbitrary and , offering a compact, low-DOF alternative suitable for high-order elliptic problems and potentially enabling VEM extensions.

Abstract

A unified construction of canonical -nonconforming finite elements is developed for -dimensional simplices for any . Consistency with the Morley-Wang-Xu elements [Math. Comp. 82 (2013), pp. 25-43] is maintained when . In the general case, the degrees of freedom and the shape function space exhibit well-matched multi-layer structures that ensure their alignment. Building on the concept of the nonconforming bubble function, the unisolvence is established using an equivalent integral-type representation of the shape function space and by applying induction on . The corresponding nonconforming finite element method applies to -th order elliptic problems, with numerical results for and in 2D supporting the theoretical analysis.
Paper Structure (19 sections, 15 theorems, 68 equations, 1 figure, 5 tables)

This paper contains 19 sections, 15 theorems, 68 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notation
  4. Review of Morley-Wang-Xu (MWX) elements
  5. Nonconforming Finite Elements
  6. Degrees of freedom
  7. Shape function space
  8. Space representation with integral terms
  9. Unisolvence
  10. Nonconforming bubbles from single-variable composition
  11. Proof of unisolvence
  12. Application to $2m$-th order elliptic PDEs
  13. Finite element space
  14. Nonconforming finite element method
  15. Error estimate
  16. ...and 4 more sections

Key Result

Lemma 2.1

If all the degrees of freedom defined in eq:MWX-DOF vanish, then for any $0\leq k \leq m$, any $(n-k)$-dimensional sub-simplex $F\in {\mathcal{F}}_{T,k}$, it holds that for any $v \in H^m(T)$, where $\nabla^j$ is the $j$th Hessian tensor for any integer $j \geq 0$.

Figures (1)

  • Figure 1: Meshes for Example 1 (left) and Example 2 (right).

Theorems & Definitions (28)

  • Lemma 2.1: Lemma 2.1 in wang2013minimal
  • Definition 2.2: weak continuity & weak zero-boundary condition
  • Remark 3.1: natural extension of MWX elements
  • Lemma 3.2: number of degrees of freedom
  • proof
  • Lemma 3.3: equivalence for vanishing DOF
  • proof : Sketch of proof
  • Remark 3.4: 1D case
  • Lemma 3.5: dimension of $P_T^{(m,n)}$
  • proof
  • ...and 18 more