Nédélec first-kind edge elements and ${\mathcal{P}}_r^{-}{\Lambda }^1$

Intuition Beginner

Some vector fields are controlled by circulation rather than by pointwise values. Electromagnetic fields are the main example: what matters along an edge or loop is the tangential component of the field.

Nédélec edge elements are finite elements built for this situation. Their degrees of freedom live on edges, and neighboring elements agree on tangential information across shared faces.

This is different from ordinary nodal elements, which match scalar values at vertices. Edge elements are designed for curl equations, just as Raviart-Thomas elements are designed for flux and divergence equations.

In FEEC, Nédélec elements are one-form elements in the discrete de Rham complex.

Visual Beginner

The basis functions are tied to oriented edges and tangential circulation.

Worked example Beginner

On a triangle, a Whitney one-form is attached to an oriented edge. Integrating the field along that edge gives its edge degree of freedom.

Nédélec first-kind elements generalize this edge-based idea to higher polynomial order and to tetrahedral meshes. The field does not need to be fully continuous as a vector field. The tangential component is the part that must match correctly.

What this tells us: conformity depends on the PDE. Curl problems need tangential conformity, not nodal scalar conformity.

Check your understanding Beginner

Formal definition Intermediate+

Let $K$ be a tetrahedron. The lowest-order Nédélec first-kind space on $K$ can be written in vector proxy notation as $$ \mathcal N_0(K)={a+b\times x:\ a,b\in\mathbb R^3}. $$ Its degrees of freedom are edge tangential moments $$ \int_e u\cdot t_e,ds $$ over every oriented edge $e$.

In FEEC notation, this is the lowest-order trimmed polynomial one-form space $$ \mathcal P_1^-\Lambda^1(K). $$ Higher-order first-kind Nédélec spaces correspond to trimmed one-form spaces ${\mathcal{P}}_r^{-}{\Lambda }^1$ under the FEEC indexing convention.

The conformity condition is membership in $$ H(\operatorname{curl};\Omega), $$ or equivalently $H{\Lambda }^1(\Omega )$ in three-dimensional form notation. Across element faces, the tangential trace must agree weakly. Full vector continuity is not required.

Counterexamples to common slips

• Nédélec edge elements are not Lagrange nodal elements applied componentwise.
• $H(\mathrm{curl})$ conformity is tangential conformity, not normal-flux conformity.
• The first-kind Nédélec family is the trimmed FEEC family; the second-kind family corresponds to a different polynomial-form family.

Key theorem with proof Intermediate+

Theorem (edge moments enforce tangential conformity). If local Nédélec fields on adjacent tetrahedra have matching tangential edge and face moments according to the element degree, their assembled field belongs to the conforming $H(\mathrm{curl})$ finite element space.

Proof. The tangential trace is the boundary quantity controlled by the $H(\mathrm{curl})$ weak formulation. On each element, a Nédélec field is polynomial and therefore has a well-defined tangential trace on every face.

The degrees of freedom are chosen so that the tangential trace on a face is determined by edge and face tangential moments. If two neighboring elements assign the same moments to their shared face with the compatible orientation convention, then their tangential traces agree as polynomial traces on that face.

When tangential traces agree across all interior faces, the distributional curl of the assembled field has no singular face contribution. Therefore the assembled field lies in $H(\mathrm{curl};\Omega )$. $\square$

Bridge. This unit extends Whitney one-forms 24.03.01 into the standard curl-conforming finite element family. It parallels Raviart-Thomas flux conformity from 24.02.02, but swaps normal flux for tangential circulation. FEEC unifies both families in 24.03.03 through polynomial differential forms.

Exercises Intermediate+

Advanced results Master

Nédélec first-kind elements are the canonical curl-conforming finite elements for Maxwell-type problems. They avoid a fundamental error made by naive nodal vector elements: electromagnetic fields need tangential trace conformity, while normal components may jump across faces.

In the lowest-order tetrahedral case, the six edge degrees of freedom match the six-dimensional space $\{a+b\times x\}$. These functions have constant plus rotational parts, and their curls are constant vector fields. This matches the first nontrivial stage of the discrete de Rham sequence.

In FEEC, the family is not defined as an isolated vector trick. It is the trimmed polynomial one-form family ${\mathcal{P}}_r^{-}{\Lambda }^1$. Its exterior derivative lands in a compatible two-form space, and this compatibility is what allows stable discretization of curl-curl equations.

The face restriction of an edge element is compatible with the lower-dimensional sequence on the face. This trace compatibility is essential for assembly over a mesh and is one reason the differential-form formulation is cleaner than a componentwise vector formulation.

The relation to Raviart-Thomas is also structural. In three dimensions, applying curl to an $H(\mathrm{curl})$ field produces an $H(\mathrm{div})$-type object in the next space of the sequence. Thus Nédélec and Raviart-Thomas spaces occupy adjacent slots in the discrete de Rham complex.

Synthesis. Nédélec elements solve the conformity problem for circulation fields. Their degrees of freedom are edge and tangential moments because one-forms integrate over curves and curl problems see tangential traces. FEEC identifies this engineering insight with the trimmed polynomial one-form space, making it part of a full complex rather than a standalone Maxwell element.

Full proof set Master

Proposition 1 (dimension of the lowest-order tetrahedral space). The space $$ \mathcal N_0(K)={a+b\times x:\ a,b\in\mathbb R^3} $$ has dimension $6$.

Proof. The vector $a$ contributes three independent constant components. The vector $b$ contributes three independent rotational components through $b\times x$. If $a+b\times x=0$ for all $x$, evaluating at $x=0$ after translating coordinates gives $a=0$, and then $b\times x=0$ for all $x$, forcing $b=0$. Thus the six parameters are independent. $\square$

Proposition 2 (edge moments match the lowest-order dimension). On a tetrahedron, the six edge tangential moments provide the correct number of degrees of freedom for ${\mathcal{N}}_0(K)$.

Proof. A tetrahedron has six edges. The lowest-order Nédélec space has dimension six by Proposition 1. The edge tangential moments are unisolvent for this space in the standard Nédélec construction: if all edge moments vanish, the tangential circulation of the field vanishes on every edge, forcing the corresponding Whitney one-form coefficients to vanish. Therefore the six moments determine the field. $\square$

Proposition 3 (curl lands in the next de Rham slot). The curl of a lowest-order Nédélec field is a constant vector proxy for a lowest-order two-form.

Proof. For $u(x)=a+b\times x$, the constant part has zero curl. The curl of $b\times x$ is a constant vector depending linearly on $b$. Hence $\mathrm{curl}u$ is constant on the element. Constant vector proxies correspond to the lowest-order face/two-form slot in the local discrete de Rham sequence. $\square$

Connections Master

• Whitney forms 24.03.01. Lowest-order Nédélec edge elements are Whitney one-forms in vector proxy language.

• Mixed FEM for Poisson 24.02.02. Raviart-Thomas elements handle normal flux; Nédélec elements handle tangential circulation.

• Sobolev spaces of differential forms 24.01.02. $H(\mathrm{curl})$ is the three-dimensional proxy for $H{\Lambda }^1$.

• Polynomial differential form spaces 24.03.03. FEEC identifies the first-kind Nédélec family with trimmed polynomial one-forms.

• Maxwell equations and FEEC edge elements 24.04.02. Maxwell discretization is the main application of Nédélec spaces.

Historical & philosophical context Master

Nédélec introduced mixed finite elements in three dimensions that became fundamental for electromagnetic computation ^[Nedelec]. Their defining feature is the correct tangential continuity for curl equations.

Bossavit and Hiptmair emphasized the geometric meaning of these elements in computational electromagnetism: degrees of freedom should live on the mesh entities over which the physical quantities are integrated ^{[Bossavit] [Hiptmair]}.

Arnold, Falk, and Winther later placed Nédélec elements into the FEEC taxonomy as trimmed polynomial differential forms ^{[Arnold-Falk-Winther]}. This showed that the element family is one piece of a broader compatible discretization of the de Rham complex.

Bibliography Master

@article{Nedelec1980,
  author = {Nedelec, Jean-Claude},
  title = {Mixed finite elements in R3},
  journal = {Numerische Mathematik},
  volume = {35},
  pages = {315--341},
  year = {1980}
}

@article{Hiptmair2002Nedelec,
  author = {Hiptmair, Ralf},
  title = {Finite elements in computational electromagnetism},
  journal = {Acta Numerica},
  volume = {11},
  pages = {237--339},
  year = {2002}
}

@book{Bossavit1998Nedelec,
  author = {Bossavit, Alain},
  title = {Computational Electromagnetism},
  publisher = {Academic Press},
  year = {1998}
}

@article{ArnoldFalkWinther2006Nedelec,
  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}
}