Smooth FEEC pointer (Falk-Neilan)

Advanced results Master

Smooth FEEC studies finite element complexes whose spaces have more continuity than the standard $H^1$, $H(\mathrm{curl})$, and $H(\mathrm{div})$ conformity levels. The motivating examples come from Stokes, biharmonic, and related systems where extra differentiability or pointwise constraints are useful.

In ordinary FEEC, the guiding object is a discrete de Rham complex. The spaces are compatible with exterior derivative, but they are not usually globally $C^1$ or smoother. Smooth FEEC asks for compatible complexes made from finite elements with higher interelement continuity.

Falk and Neilan's 2013 work constructs Stokes complexes that support stable finite elements with pointwise mass conservation. The pointwise divergence-free feature is stronger than a weak incompressibility statement. It lets the discrete velocity satisfy an exact local conservation law, a property that matters in fluid computation and in pressure-robust discretization.

The technical challenge is that high continuity and exactness pull against each other. Enforcing $C^1$-type continuity across triangles or tetrahedra requires additional vertex, edge, and macroelement constraints. At the same time, the derivative maps must still fit into a complex with the correct kernels and images.

Neilan's three-dimensional extension shows that the smooth-complex program is not a two-dimensional accident. The construction uses carefully designed polynomial spaces and degrees of freedom so that the complex is conforming and exact at the discrete level.

For this curriculum, smooth FEEC is best treated as a pointer rather than a prerequisite. The core conceptual lesson is already present in 24.03.04 through 24.03.06: stability comes from compatible complexes and commuting projections. Smooth FEEC adds the extra design constraint of higher continuity.

The main application class is incompressible flow and related mixed systems. In Stokes flow, exact divergence-free velocities can improve mass conservation and remove certain pressure-dependence artifacts in the velocity error. Smooth complexes give a systematic route to such elements.

Synthesis. Smooth FEEC extends the FEEC principle into high-continuity finite element design. It keeps the same core structure, a complex plus stable projections, but moves into element families where smoothness, exactness, and conservation must be engineered together.

Connections Master

• Discrete de Rham complex 24.03.04. Smooth FEEC keeps the complex requirement but raises continuity.

• Bounded cochain projection 24.03.05. Smooth complexes still need stable commuting projections or Fortin-style operators.

• FEEC convergence theorem 24.03.06. The stability philosophy is inherited from the main FEEC convergence mechanism.

• Mixed Hodge Laplacian 24.04.01. Smooth FEEC can be read as a higher-continuity extension of compatible mixed discretization.

• Linearised elasticity 24.04.03. Both elasticity and smooth FEEC show that important PDE complexes go beyond the lowest-order de Rham examples.

Historical & philosophical context Master

The standard FEEC story unified many stable finite element families, but it did not exhaust the design space. Higher-continuity methods were already important for plate equations, biharmonic problems, and incompressible flow.

Falk and Neilan's 2013 paper connected Stokes complexes to the construction of stable elements with pointwise mass conservation ^{[Falk-Neilan]}. Neilan's 2015 work extended the construction to conforming smooth de Rham complexes in three dimensions ^[Neilan].

The philosophical point is that compatibility is a structural design law, not a restriction to low-continuity spaces. Once the PDE asks for a complex, the finite element designer can pursue smoothness, local conservation, or special constraints as long as the complex structure survives.

Bibliography Master

@article{FalkNeilan2013SmoothFEEC,
  author = {Falk, Richard S. and Neilan, Michael},
  title = {Stokes complexes and the construction of stable finite elements with pointwise mass conservation},
  journal = {SIAM Journal on Numerical Analysis},
  volume = {51},
  pages = {1308--1326},
  year = {2013}
}

@article{Neilan2015SmoothFEEC,
  author = {Neilan, Michael},
  title = {Discrete and conforming smooth de Rham complexes in three dimensions},
  journal = {Mathematics of Computation},
  volume = {84},
  pages = {2059--2081},
  year = {2015}
}

@article{ArnoldFalkWinther2006SmoothPointer,
  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}
}