Discrete de Rham complex and the FEEC subcomplex axiom

Intuition Beginner

A good finite element method should not break the structure of the differential operators.

For scalar, vector, and flux fields, the continuous chain is gradient, then curl, then divergence. Applying the next operator after the previous one gives zero. This is not a coincidence; it is the de Rham complex in vector-calculus language.

The discrete de Rham complex asks finite element spaces to follow the same pattern. If a discrete scalar field is differentiated, the result should land in the discrete edge-field space. If that is differentiated, it should land in the discrete face-field space.

This is the FEEC subcomplex axiom: the discrete spaces must form a smaller version of the continuous complex.

Visual Beginner

The bottom row is a finite-dimensional chain that sits inside the continuous row and respects the same derivative arrows.

Worked example Beginner

Whitney forms give the simplest example. Vertex functions differentiate into edge forms. Edge forms differentiate into face forms. Face forms differentiate into cell volume forms.

This mirrors the mesh topology: vertices, edges, faces, and cells are connected by incidence relations.

What this tells us: a finite element complex is not just a collection of approximation spaces. It is a compatible chain of spaces.

Check your understanding Beginner

Formal definition Intermediate+

Let $$ 0\to H\Lambda^0(\Omega)\xrightarrow{d}H\Lambda^1(\Omega)\xrightarrow{d}\cdots\xrightarrow{d}H\Lambda^n(\Omega)\to0 $$ be the continuous de Rham Hilbert complex.

A finite element de Rham subcomplex is a family of finite-dimensional spaces $$ \Lambda_h^k\subset H\Lambda^k(\Omega) $$ such that $$ d\Lambda_h^k\subset \Lambda_h^{k+1} $$ for every $k$. Thus $$ 0\to \Lambda_h^0\xrightarrow{d}\Lambda_h^1\xrightarrow{d}\cdots\xrightarrow{d}\Lambda_h^n\to0 $$ is a cochain complex.

On simplicial meshes, FEEC supplies two canonical polynomial-form subcomplexes: $$ \mathcal P_r\Lambda^0 \xrightarrow{d} \mathcal P_{r-1}\Lambda^1 \xrightarrow{d} \cdots \xrightarrow{d} \mathcal P_{r-n}\Lambda^n, $$ where indices are interpreted only when nonnegative, and the trimmed sequence $$ \mathcal P_r^-\Lambda^0 \xrightarrow{d} \mathcal P_r^-\Lambda^1 \xrightarrow{d} \cdots \xrightarrow{d} \mathcal P_r^-\Lambda^n. $$

The cohomology of the discrete complex is $$ H_h^k=\ker(d:\Lambda_h^k\to\Lambda_h^{k+1})/ d\Lambda_h^{k-1}. $$ A good FEEC complex preserves the relevant cohomology of the domain, typically matching simplicial or de Rham cohomology under appropriate assumptions.

Counterexamples to common slips

• A collection of finite element spaces is not a FEEC complex unless $d$ maps each space into the next.
• Approximation quality alone is not enough; the complex structure is part of the stability mechanism.
• Discrete cohomology is not a numerical artifact to ignore. It records the topology that the method must represent.

Key theorem with proof Intermediate+

Theorem (FEEC subcomplex property for polynomial forms). The full and trimmed FEEC polynomial families assemble into finite-dimensional subcomplexes of the continuous de Rham complex on a conforming simplicial mesh.

Proof. On each simplex, 24.03.03 gives exterior-derivative compatibility for the local polynomial form spaces. For the full family, $d$ maps the chosen polynomial degree in form degree $k$ into the chosen degree in form degree $k+1$. For the trimmed family, $d$ preserves the trimmed family index.

The trace compatibility of polynomial differential forms ensures that when local forms are assembled across shared faces, the exterior derivative of the assembled form is the assembly of the local exterior derivatives. Thus no incompatible face terms appear.

Therefore, if ${\omega }_h\in {\Lambda }_h^k$, then $d{\omega }_h$ belongs to the next global finite element space ${\Lambda }_h^{k+1}$. Since $d^2=0$ already holds for differential forms, the discrete sequence is a cochain complex. $\square$

Bridge. This unit uses Whitney forms 24.03.01 and polynomial differential-form spaces 24.03.03 to state the structural FEEC axiom. It prepares 24.03.05, where bounded cochain projections make the diagram commute, and 24.03.06, where the subcomplex plus projection gives stability and convergence.

Exercises Intermediate+

Advanced results Master

The discrete de Rham complex is the algebraic backbone of FEEC. Its role is analogous to choosing a conforming scalar space in classical FEM, but stronger: one chooses a whole compatible family of spaces, one for each form degree.

The subcomplex axiom ensures that differentiation does not leave the finite element sequence. This makes the discrete problem inherit identities from the continuous complex, especially $d^2=0$. In vector proxy language, this preserves $\mathrm{curl}\nabla =0$ and $\mathrm{div}\mathrm{curl}=0$.

There are two standard FEEC sequences on simplicial meshes: a full sequence and a trimmed sequence. The full sequence shifts polynomial degree as form degree rises, while the trimmed sequence keeps a fixed FEEC index. These two sequences recover the classical element families in different slots.

Discrete cohomology is controlled by the mesh topology. With the right finite element complex and maps to simplicial cochains, the discrete cohomology matches the cohomology of the underlying triangulated domain. This is crucial for harmonic forms and Hodge Laplacian problems.

The subcomplex axiom by itself is not the whole FEEC stability theorem. It must be paired with bounded cochain projections, developed in 24.03.05. The subcomplex says the spaces fit together; the projection says the continuous and discrete complexes communicate stably.

Synthesis. The FEEC subcomplex axiom is the point where compatible finite elements become a homological construction. The finite element spaces are not chosen one PDE at a time; they form a discrete de Rham complex that preserves derivative identities and topology. This is the structural reason FEEC can prove stability results uniformly across scalar, curl, divergence, and Hodge-Laplacian problems.

Full proof set Master

Proposition 1 (discrete sequence is a complex). If $d{\Lambda }_h^k\subset {\Lambda }_h^{k+1}$ for all $k$, then the finite element sequence is a cochain complex.

Proof. The maps are the restrictions of the exterior derivative to the finite element spaces. The subcomplex condition makes each restricted map well-defined. Since the exterior derivative satisfies $d^2=0$ on the continuous complex, its restrictions also satisfy $d^2=0$. Therefore the finite element sequence is a cochain complex. $\square$

Proposition 2 (Whitney forms give a lowest-order subcomplex). The Whitney spaces form a discrete de Rham subcomplex.

Proof. The Whitney map satisfies $dW=W\delta$ by 24.03.01. Thus differentiating a Whitney form gives the Whitney form of a simplicial coboundary, which lies in the next Whitney space. Hence $d{\Lambda }_h^k\subset {\Lambda }_h^{k+1}$ for the Whitney sequence. $\square$

Proposition 3 (subcomplex preserves derivative identities). In a three-dimensional vector proxy sequence, the discrete operators satisfy the analogues of $\mathrm{curl}\nabla =0$ and $\mathrm{div}\mathrm{curl}=0$.

Proof. The vector proxy sequence is the de Rham sequence in degrees $0$ through $3$. The identities $\mathrm{curl}\nabla =0$ and $\mathrm{div}\mathrm{curl}=0$ are the degree-specific forms of $d^2=0$. Since the discrete spaces form a subcomplex under the same exterior derivative, the restricted operators also satisfy these identities. $\square$

Connections Master

• Whitney forms 24.03.01. Whitney forms give the lowest-order example of a discrete de Rham complex.

• Polynomial differential form spaces 24.03.03. Full and trimmed polynomial spaces supply the standard FEEC subcomplexes.

• De Rham cohomology 03.04.06. Discrete cohomology mirrors the continuous de Rham cohomology of the domain.

• Bounded cochain projection 24.03.05. A projection commuting with $d$ is the next structural axiom needed for stability.

• FEEC convergence theorem 24.03.06. The convergence theorem uses the subcomplex and bounded projection together.

Historical & philosophical context Master

The de Rham complex began as a bridge between differential forms and topology. Whitney's work connected differential forms to cochains on triangulations, giving a geometric route from smooth forms to discrete topology ^[Whitney].

Arnold, Falk, and Winther used this bridge as the foundation of FEEC ^{[Arnold-Falk-Winther]}. Their insight was that many finite element stability problems are failures or successes of discrete complex structure.

The subcomplex axiom expresses a philosophical shift in numerical PDE: a stable discretization should preserve the algebraic identities of the differential operators, not merely approximate each operator separately.

Bibliography Master

@article{ArnoldFalkWinther2006DiscreteComplex,
  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}
}

@article{ArnoldFalkWinther2010FEEC,
  author = {Arnold, Douglas N. and Falk, Richard S. and Winther, Ragnar},
  title = {Finite element exterior calculus: from Hodge theory to numerical stability},
  journal = {Bulletin of the American Mathematical Society},
  volume = {47},
  pages = {281--354},
  year = {2010}
}

@book{Arnold2018DiscreteComplex,
  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{Whitney1957DiscreteComplex,
  author = {Whitney, Hassler},
  title = {Geometric Integration Theory},
  publisher = {Princeton University Press},
  year = {1957}
}