Maxwell equations and FEEC edge elements

Intuition Beginner

Maxwell's equations describe electric and magnetic fields. In computation, the most delicate part is not drawing a field on a mesh. The delicate part is choosing mesh variables that match the way electromagnetic fields really fit together.

Electric field circulation is tied to edges and loops. That is why edge elements are the natural finite elements for many Maxwell problems. They record tangential information along edges rather than only values at vertices.

This choice prevents a common numerical failure: false electromagnetic modes that appear because the mesh method has the wrong field geometry.

FEEC explains the cure. Maxwell's equations sit inside a chain of compatible derivative operations. A good discretization preserves that chain on the mesh.

Visual Beginner

The edge field represents circulation, and the face field represents the next derivative stage.

Worked example Beginner

Imagine simulating the electric field in a cavity. If the method stores only vertex values for each vector component, it may force the wrong kind of continuity across element faces.

An edge-element method instead stores circulation along mesh edges. Neighboring tetrahedra share the same tangential information on common faces, which matches the curl equation.

What this tells us: Maxwell discretization is not just approximation. It is geometry preservation.

Check your understanding Beginner

Formal definition Intermediate+

Let $\Omega \subset {\mathbb{R}}^3$ be a bounded domain with material coefficients $\mu$ and $\epsilon$. A standard time-harmonic electric-field formulation seeks $$ E\in H_0(\operatorname{curl};\Omega) $$ such that $$ \operatorname{curl},\mu^{-1}\operatorname{curl}E-\omega^2\varepsilon E=i\omega J $$ in weak form. The Galerkin form is $$ (\mu^{-1}\operatorname{curl}E,\operatorname{curl}F) -\omega^2(\varepsilon E,F) =(i\omega J,F) $$ for all test fields $F\in H_0(\mathrm{curl};\Omega )$, with boundary conditions encoded by the trace space.

The FEEC reading identifies $$ H(\operatorname{curl};\Omega)\simeq H\Lambda^1(\Omega) $$ in three-dimensional vector proxy notation. The curl is the exterior derivative from one-forms to two-forms.

The finite element replacement is a Nedelec space $$ V_h^1\subset H(\operatorname{curl};\Omega), $$ usually part of a discrete de Rham sequence $$ V_h^0 \xrightarrow{\nabla} V_h^1 \xrightarrow{\operatorname{curl}} V_h^2 \xrightarrow{\operatorname{div}} V_h^3. $$ For first-kind edge elements, $V_h^1$ is the vector-proxy version of a trimmed polynomial one-form space from 24.03.02 and 24.03.03.

Counterexamples to common slips

• Maxwell edge elements are not componentwise scalar Lagrange elements.
• The gradient kernel is not numerical noise; it is part of the de Rham complex.
• Spurious eigenmodes are not fixed only by raising polynomial degree; the discrete complex and projection structure matter.

Key theorem with proof Intermediate+

Theorem (FEEC Maxwell stability pattern). Suppose the Maxwell finite element spaces form a discrete de Rham subcomplex and admit mesh-uniform bounded cochain projections. Then Nedelec Galerkin discretizations of the well-posed time-harmonic Maxwell problem preserve the gradient-kernel structure and converge under the FEEC approximation hypotheses. For cavity eigenvalue problems, the same complex structure is the mechanism behind the absence of spurious modes.

Proof. The continuous Maxwell curl-curl operator is not an isolated vector operator. It sits in the de Rham complex. The kernel contains gradient fields, and the physically relevant part is controlled on a complement of that kernel by a Maxwell/Poincare estimate.

The Nedelec space $V_h^1$ is chosen so that $\mathrm{curl}{V}_h^1\subset V_h^2$. The discrete gradient space from $V_h^0$ lands inside $V_h^1$, so the identity $\mathrm{curl}\nabla =0$ is preserved exactly on the mesh.

The bounded cochain projection from 24.03.05 transfers continuous estimates to the discrete complex. This prevents mesh-dependent kernel pollution and gives the discrete compactness/Poincare control used in Maxwell analysis.

Once stability is established on the appropriate complement, Galerkin orthogonality and polynomial approximation give convergence. In the eigenvalue setting, the same compactness mechanism rules out spectral branches that do not approximate genuine Maxwell modes. $\square$

Bridge. This unit applies the Nedelec edge-element construction of 24.03.02 and the FEEC convergence mechanism of 24.03.06 to Maxwell equations. It is the electromagnetic counterpart to the mixed Hodge Laplacian model problem 24.04.01.

Exercises Intermediate+

Advanced results Master

The time-harmonic Maxwell operator is a curl-curl operator with a mass term. At zero frequency or at resonant frequencies, kernel and compatibility issues become central. Even away from resonance, the finite element method must preserve the de Rham structure to avoid polluting the spectrum.

The Nedelec space is the degree-one slot of the discrete de Rham complex. Its tangential trace is the natural boundary trace for electric fields, while the curl lands in a face/flux-type space. This is why edge elements and Raviart-Thomas-type spaces appear next to each other in electromagnetic computation.

Hiptmair's Acta Numerica treatment places computational electromagnetism in exactly this complex-preserving framework: degrees of freedom live on the mesh entities over which the physical field quantities are integrated. Electric circulation belongs on edges, magnetic flux belongs on faces, charge density belongs in cells, and potentials belong at vertices.

FEEC adds a uniform theorem structure. The subcomplex condition preserves algebraic identities such as $d^2=0$. Bounded cochain projections make the continuous/discrete comparison stable. Together they give the compactness and approximation estimates that underpin convergence and spectral correctness.

For eigenvalue computations in cavities, this matters directly. Noncompatible vector elements can produce eigenvalues that refine to no true electromagnetic mode. Edge elements in a stable de Rham complex suppress that failure by preserving the gradient kernel and controlling the orthogonal complement.

Synthesis. Maxwell FEEC is the meeting point of physical field geometry and numerical stability. Maxwell supplies the curl-curl operator, Nedelec supplies the tangentially conforming edge space, and FEEC explains why the whole de Rham sequence must be discretized together.

Full proof set Master

Proposition 1 (the Maxwell curl-curl operator is a degree-one complex operator). In three-dimensional vector proxy notation, the electric-field curl-curl operator acts on the one-form slot of the de Rham complex.

Proof. A vector electric field with tangential trace corresponds to a one-form in $H{\Lambda }^1(\Omega )$. The exterior derivative sends one-forms to two-forms, and in vector proxy notation this is curl. Applying the adjoint/material-weighted return map and curl-curl weak form gives the Maxwell operator on the one-form slot. $\square$

Proposition 2 (compatible discretization preserves the gradient kernel). If $$ V_h^0 \xrightarrow{\nabla} V_h^1 \xrightarrow{\operatorname{curl}} V_h^2 $$ is a subcomplex, then $\nabla V_h^0\subset \ker (\mathrm{curl}{|}_{V_h^1})$.

Proof. The subcomplex condition says the derivative maps one discrete space into the next. Since the continuous identity $\mathrm{curl}\nabla =0$ holds polynomially on each element and the traces assemble compatibly, every discrete gradient has zero discrete curl. Therefore the image of $V_h^0$ lies inside the discrete curl kernel. $\square$

Proposition 3 (why componentwise nodal vector elements can fail). A vector-valued nodal Lagrange space is not, by itself, a Maxwell-compatible FEEC space unless it participates in a stable discrete de Rham complex with the correct traces.

Proof. Maxwell's weak form requires conformity in $H(\mathrm{curl})$, whose natural trace is tangential. Componentwise nodal continuity imposes a different condition and does not automatically supply the exact sequence, cochain projection, or discrete compactness needed for Maxwell spectral correctness. Without those structures, the discrete kernel and complement can mismatch the continuous ones, creating spurious modes. $\square$

Connections Master

• Nedelec edge elements 24.03.02. Maxwell is the main application of curl-conforming edge elements.

• Discrete de Rham complex 24.03.04. The Maxwell method uses the degree-one slot and its neighboring gradient and curl maps.

• Bounded cochain projection 24.03.05. Stable commuting projection supplies the discrete compactness mechanism.

• FEEC convergence theorem 24.03.06. Maxwell inherits FEEC's stability pattern for compatible Hilbert complexes.

• Mixed Hodge Laplacian 24.04.01. The Hodge Laplacian is the model FEEC problem; Maxwell is a physically central curl-curl specialization.

• Maxwell's equations in differential form 10.04.01. The physics unit supplies the differential-form statement of electromagnetic field equations.

Historical & philosophical context Master

Maxwell's 1865 paper unified electricity, magnetism, and light into a field theory ^[Maxwell]. The later numerical question is how to compute this field theory without breaking its geometric identities.

Nedelec's 1980 paper introduced three-dimensional mixed finite elements whose tangential continuity made them suitable for electromagnetic curl problems ^[Nedelec]. Bossavit and Hiptmair developed the computational electromagnetism viewpoint in which mesh degrees of freedom are assigned to geometric entities that match the field quantities ^{[Bossavit] [Hiptmair]}.

FEEC folds this history into a general mathematical framework. The message is that Maxwell computation is not a special vector-calculus trick. It is the degree-one case of compatible discretization of the de Rham complex.

Bibliography Master

@article{Maxwell1865DynamicalTheory,
  author = {Maxwell, James Clerk},
  title = {A Dynamical Theory of the Electromagnetic Field},
  journal = {Philosophical Transactions of the Royal Society of London},
  volume = {155},
  pages = {459--512},
  year = {1865}
}

@article{Nedelec1980MaxwellFEEC,
  author = {Nedelec, Jean-Claude},
  title = {Mixed finite elements in R3},
  journal = {Numerische Mathematik},
  volume = {35},
  pages = {315--341},
  year = {1980}
}

@article{Hiptmair2002MaxwellFEEC,
  author = {Hiptmair, Ralf},
  title = {Finite elements in computational electromagnetism},
  journal = {Acta Numerica},
  volume = {11},
  pages = {237--339},
  year = {2002}
}

@book{Bossavit1998MaxwellFEEC,
  author = {Bossavit, Alain},
  title = {Computational Electromagnetism},
  publisher = {Academic Press},
  year = {1998}
}

@article{ArnoldFalkWinther2006MaxwellFEEC,
  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}
}