Polynomial differential form spaces ${\mathcal{P}}_r{\Lambda }^k$ and ${\mathcal{P}}_r^{-}{\Lambda }^k$

Intuition Beginner

FEEC organizes finite elements by two labels: the degree of the differential form and the degree of the polynomial.

The form degree says what kind of geometric quantity is being approximated: vertex values, edge circulations, face fluxes, or cell densities.

The polynomial degree says how rich the local approximation is inside each mesh element.

Classical finite element families that look separate at first become parts of one table. Lagrange, Raviart-Thomas, Brezzi-Douglas-Marini, and Nédélec elements are all polynomial differential-form spaces viewed in different degrees.

Visual Beginner

FEEC turns many named finite element families into entries in one structured table.

Worked example Beginner

Whitney forms are the lowest-order row of this picture. They give vertex, edge, face, and cell basis functions compatible with the mesh topology.

Increasing the polynomial degree enriches the local approximation. Instead of only the simplest edge or face behavior, the element can represent more variation inside each simplex.

What this tells us: FEEC is not a single element. It is a systematic way to generate compatible families of elements.

Check your understanding Beginner

Formal definition Intermediate+

Let $K$ be an $n$-simplex. The space ${\mathcal{P}}_r{\Lambda }^k(K)$ consists of differential $k$-forms on $K$ whose coefficient functions are polynomials of degree at most $r$.

The trimmed space ${\mathcal{P}}_r^{-}{\Lambda }^k(K)$ is the smaller FEEC family defined by $$ \mathcal P_r^-\Lambda^k(K) =\mathcal P_{r-1}\Lambda^k(K)+\kappa\mathcal P_{r-1}\Lambda^{k+1}(K), $$ where $\kappa$ is the Koszul operator, contraction with the position vector field.

The two families satisfy the important nesting relation $$ \mathcal P_r^-\Lambda^k(K)\subset \mathcal P_r\Lambda^k(K)\subset \mathcal P_{r+1}^-\Lambda^k(K). $$

The exterior derivative maps these spaces in compatible ways: $$ d:\mathcal P_r\Lambda^k\to\mathcal P_{r-1}\Lambda^{k+1}, $$ and $$ d:\mathcal P_r^-\Lambda^k\to\mathcal P_r^-\Lambda^{k+1}. $$

In vector proxy language, these spaces recover familiar elements:

• ${\mathcal{P}}_r{\Lambda }^0$: Lagrange-type scalar polynomial spaces.
• ${\mathcal{P}}_r^{-}{\Lambda }^1$: Nédélec first-kind edge elements.
• ${\mathcal{P}}_r{\Lambda }^1$: Nédélec second-kind edge elements.
• ${\mathcal{P}}_r^{-}{\Lambda }^{n-1}$: Raviart-Thomas-type flux elements.
• ${\mathcal{P}}_r{\Lambda }^{n-1}$: Brezzi-Douglas-Marini-type flux elements.

Counterexamples to common slips

• The trimmed space is not merely "lower degree everywhere"; it includes a Koszul term that preserves the complex structure.
• The full and trimmed families are different but interleaved by degree.
• The exterior derivative lowers polynomial degree for the full family but preserves the trimmed-family index in the FEEC convention.

Key theorem with proof Intermediate+

Theorem (polynomial form spaces form compatible families). The full and trimmed polynomial form families are compatible with the exterior derivative: $$ d\mathcal P_r\Lambda^k\subset \mathcal P_{r-1}\Lambda^{k+1}, \qquad d\mathcal P_r^-\Lambda^k\subset \mathcal P_r^-\Lambda^{k+1}. $$

Proof. For the full family, a polynomial $k$-form in ${\mathcal{P}}_r{\Lambda }^k$ has coefficient polynomials of degree at most $r$. Applying $d$ differentiates those coefficients, so the resulting coefficient polynomials have degree at most $r-1$. Thus $$ d\mathcal P_r\Lambda^k\subset \mathcal P_{r-1}\Lambda^{k+1}. $$

For the trimmed family, write an element as $$ \omega=\alpha+\kappa\beta, $$ with $\alpha \in {\mathcal{P}}_{r-1}{\Lambda }^k$ and $\beta \in {\mathcal{P}}_{r-1}{\Lambda }^{k+1}$. The Koszul operator satisfies homotopy identities with $d$ on homogeneous polynomial forms. These identities show that $d(\kappa \beta )$ decomposes into a polynomial form part plus another Koszul-controlled part of the correct trimmed degree. Hence $d\omega \in {\mathcal{P}}_r^{-}{\Lambda }^{k+1}$. $\square$

Bridge. This unit is the algebraic kernel behind the element examples in 24.02.02, 24.03.01, and 24.03.02. It prepares 24.03.04, where these spaces are assembled into a discrete de Rham complex, and 24.03.05, where projections must commute with the exterior derivative.

Exercises Intermediate+

Advanced results Master

The Koszul operator is contraction with the vector field pointing from the origin of the simplex coordinates. Algebraically, it is the companion to the exterior derivative in a polynomial de Rham complex. Together, $d$ and $\kappa$ organize homogeneous polynomial forms by degree and form degree.

The full family ${\mathcal{P}}_r{\Lambda }^k$ and trimmed family ${\mathcal{P}}_r^{-}{\Lambda }^k$ are the two canonical FEEC families on simplices. Their traces on faces remain in the corresponding spaces on the faces, which is essential for assembling global conforming spaces.

The full family gives the second-kind sequence; the trimmed family gives the first-kind sequence. In low degrees and dimensions, the names translate into the classical element table: Lagrange, Nédélec first kind, Nédélec second kind, Raviart-Thomas, Brezzi-Douglas-Marini, and discontinuous cell polynomials.

Dimension formulas come from counting polynomial $k$-form coefficients and subtracting or adding the Koszul-controlled pieces. The point of these formulas is not only bookkeeping. Matching dimensions with degrees of freedom is what gives unisolvence, and unisolvence is what makes local finite elements usable in assembly.

The corrected inclusion chain $$ \mathcal P_r^-\Lambda^k\subset \mathcal P_r\Lambda^k\subset \mathcal P_{r+1}^-\Lambda^k $$ is often the easiest way to remember the relationship between the two families: trimmed spaces are slightly smaller at the same index, while the next trimmed level contains the full one.

Synthesis. Polynomial differential forms are FEEC's classification engine. Instead of memorizing separate finite element families, one studies two algebraic families indexed by polynomial degree and form degree. The Koszul operator creates the trimmed family, the exterior derivative links adjacent form degrees, and the classical named elements appear as shadows of this uniform construction.

Full proof set Master

Proposition 1 (full polynomial derivative degree). If $\omega \in {\mathcal{P}}_r{\Lambda }^k(K)$, then $d\omega \in {\mathcal{P}}_{r-1}{\Lambda }^{k+1}(K)$.

Proof. Write $\omega$ as a finite sum of polynomial coefficients times basis $k$-forms. The exterior derivative differentiates the coefficient polynomials and wedges with coordinate one-forms. Differentiating a polynomial of degree at most $r$ gives a polynomial of degree at most $r-1$. Hence $d\omega$ has coefficients of degree at most $r-1$. $\square$

Proposition 2 (Whitney forms are lowest-order trimmed forms). Whitney $k$-forms on a simplex lie in the lowest-order trimmed polynomial $k$-form space.

Proof. A Whitney $k$-form is built from barycentric coordinate functions and differentials of barycentric coordinates. The barycentric coordinates are affine polynomials, and their differentials are constant one-forms. The alternating barycentric expression has exactly the polynomial structure of the lowest-order trimmed family. Therefore the Whitney space is the lowest-order trimmed space in degree $k$. $\square$

Proposition 3 (first-kind edge elements are trimmed one-forms). The Nédélec first-kind edge element family is the vector proxy of ${\mathcal{P}}_r^{-}{\Lambda }^1$.

Proof. In three dimensions, one-forms correspond to vector fields through the Euclidean proxy. The degrees of freedom of ${\mathcal{P}}_r^{-}{\Lambda }^1$ are tangential moments on edges and higher-dimensional subsimplices, matching the Nédélec first-kind degrees of freedom. The polynomial content is the trimmed one-form content, which in vector notation gives the classical first-kind Nédélec shape functions. Thus the families agree under the proxy. $\square$

Connections Master

• Whitney forms 24.03.01. Whitney forms are the lowest-order instance of the trimmed polynomial form family.

• Nédélec first-kind edge elements 24.03.02. These are ${\mathcal{P}}_r^{-}{\Lambda }^1$ spaces in three-dimensional vector proxy language.

• Mixed FEM for Poisson 24.02.02. Raviart-Thomas and BDM flux elements appear as polynomial form spaces in degree $n-1$.

• Exterior derivative 03.04.04. FEEC polynomial spaces are designed around how $d$ maps between form degrees.

• Discrete de Rham complex 24.03.04. These polynomial spaces become the finite-dimensional subcomplexes used in FEEC.

Historical & philosophical context Master

Classical finite element theory developed many families for different PDEs: nodal elements for scalar elliptic problems, Raviart-Thomas and BDM elements for fluxes, and Nédélec elements for electromagnetism.

Arnold, Falk, and Winther's FEEC synthesis showed that these families are organized by polynomial differential forms and homological algebra ^{[Arnold-Falk-Winther]}. The Koszul operator and trimmed spaces are the algebraic devices that make the classification work.

Arnold's CBMS lectures present this as the mathematical kernel of FEEC: once the polynomial form families are understood, the later discrete de Rham complex, commuting projections, and convergence theorem become structural rather than case-by-case ^[Arnold].

Bibliography Master

@article{ArnoldFalkWinther2006PolynomialForms,
  author = {Arnold, Douglas N. and Falk, Richard S. and Winther, Ragnar},
  title = {Finite element exterior calculus, homological techniques, and applications},
  journal = {Acta Numerica},
  volume = {15},
  pages = {1--155},
  year = {2006}
}

@book{Arnold2018PolynomialForms,
  author = {Arnold, Douglas N.},
  title = {Finite Element Exterior Calculus},
  series = {CBMS-NSF Regional Conference Series in Applied Mathematics},
  volume = {93},
  publisher = {SIAM},
  year = {2018}
}

@book{BoffiBrezziFortin2013PolynomialForms,
  author = {Boffi, Daniele and Brezzi, Franco and Fortin, Michel},
  title = {Mixed Finite Element Methods and Applications},
  series = {Springer Series in Computational Mathematics},
  volume = {44},
  publisher = {Springer},
  year = {2013}
}

@article{Nedelec1980PolynomialForms,
  author = {Nedelec, Jean-Claude},
  title = {Mixed finite elements in R3},
  journal = {Numerische Mathematik},
  volume = {35},
  pages = {315--341},
  year = {1980}
}