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Finite element spaces of double forms

Yakov Berchenko-Kogan, Evan S. Gawlik

Abstract

The tensor product of two differential forms of degree $p$ and $q$ is a multilinear form that is alternating in its first $p$ arguments and alternating in its last $q$ arguments. These forms, which are known as double forms or $(p,q)$-forms, play a central role in certain differential complexes that arise when studying partial differential equations. We construct piecewise polynomial finite element spaces for all of the natural subspaces of the space of $(p,q)$-forms, excluding one subspace which fails to admit a piecewise constant discretization. As special cases, our construction recovers known finite element spaces for symmetric matrices with tangential-tangential continuity (the Regge finite elements), symmetric matrices with normal-normal continuity, and trace-free matrices with normal-tangential continuity. It also gives rise to new spaces, like a finite element space for tensors possessing the symmetries of the Riemann curvature tensor.

Finite element spaces of double forms

Abstract

The tensor product of two differential forms of degree and is a multilinear form that is alternating in its first arguments and alternating in its last arguments. These forms, which are known as double forms or -forms, play a central role in certain differential complexes that arise when studying partial differential equations. We construct piecewise polynomial finite element spaces for all of the natural subspaces of the space of -forms, excluding one subspace which fails to admit a piecewise constant discretization. As special cases, our construction recovers known finite element spaces for symmetric matrices with tangential-tangential continuity (the Regge finite elements), symmetric matrices with normal-normal continuity, and trace-free matrices with normal-tangential continuity. It also gives rise to new spaces, like a finite element space for tensors possessing the symmetries of the Riemann curvature tensor.

Paper Structure

This paper contains 41 sections, 79 theorems, 173 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Organization
  3. Related work
  4. Double multicovectors
  5. The $s$ and $s^*$ operators
  6. The decomposition of double multicovectors
  7. Properties of the decomposition
  8. Decomposition examples
  9. The $m=q$ summand
  10. The decomposition for $(p,q)=(1,1)$.
  11. The decomposition for $(p,q)=(2,1)$.
  12. The decomposition for $(p,q)=(2,2)$.
  13. Symmetric and skew-symmetric $(p,p)$-forms.
  14. Additional operations on double multicovectors
  15. Double forms
  16. ...and 26 more sections

Key Result

Proposition 2.7

We have where $\alpha$ is a $p$-covector and $\beta$ is a $q$-covector.

Figures (7)

  • Figure 1: A form which vanishes on vectors tangent to dilates of $\partial T^n$ need not vanish on all vectors tangent to the coordinate hyperplanes.
  • Figure 2: The Young diagram for $\Lambda^{p,q}_m$
  • Figure 3: Tensor indices associated to each box of the Young diagram of $\Lambda^{p,q}_0$
  • Figure 4: The numbers $n-i+j$ (left) and the hook lengths (right) for the Young diagram of $\Lambda^{p,q}_0$
  • Figure 5: The Young diagram for the image of $d_Ld_R\colon\mathcal{H}_{r+2}\Lambda^{p-1,q-1}_0\to\mathcal{H}_r\Lambda^{p,q}_0$
  • ...and 2 more figures

Theorems & Definitions (194)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • proof
  • Lemma 2.9
  • ...and 184 more