Virtual element exterior calculus pointer

Advanced results Master

Virtual element exterior calculus extends FEEC ideas to polygonal and polyhedral meshes. The virtual element method allows local spaces whose basis functions are not written down explicitly inside each element, as long as the degrees of freedom and projection operators needed for computation are available.

This is valuable when the mesh has general cell shapes. Standard finite elements are easiest on triangles, tetrahedra, quadrilaterals, and hexahedra. Many geometric and engineering workflows produce cells with arbitrary numbers of faces, cut cells, nonmatching interfaces, or polyhedral aggregates. VEM is designed for that setting.

Exterior calculus enters when the virtual spaces are organized into a compatible de Rham complex. One builds virtual scalar, edge, face, and volume spaces so that the discrete derivative maps land in the next space and the relevant projections commute.

The word virtual signals a computational distinction. The full basis functions may not be evaluated pointwise in the cell interior. Instead, one computes polynomial projections, boundary traces, and stabilization terms from degrees of freedom. The method is engineered so that all bilinear forms needed by the PDE are computable.

Beirao da Veiga, Brezzi, Marini, Russo, and collaborators developed the VEM framework and its compatible variants. The 2018 construction cited in the FEEC audit points toward virtual element exterior calculus on polygonal and polyhedral meshes.

For the curriculum, this is a modern landscape pointer. The core FEEC units already explain why complexes, projections, and cohomology matter. VEM exterior calculus shows that the same requirements can be met even when classical element shape functions are replaced by implicit local spaces.

The main tradeoff is design complexity. Virtual spaces give mesh flexibility, but the analyst must prove unisolvence, projection computability, stability of the stabilization term, approximation estimates, and exactness or cohomology control for the virtual complex.

Synthesis. Virtual element exterior calculus keeps the FEEC compatibility principle while relaxing the geometry of the mesh and the explicitness of local basis functions. It is the compatible-method route for general polygonal and polyhedral meshes.

Connections Master

• Discrete de Rham complex 24.03.04. Virtual exterior calculus still depends on a discrete complex.

• Bounded cochain projection 24.03.05. Projection computability and commuting properties are central to VEM stability.

• FEEC convergence theorem 24.03.06. The convergence proof pattern is inherited from compatible complex methods.

• Isogeometric exterior calculus 24.04.05. Both generalize FEEC beyond standard simplex polynomial elements, but in different directions.

• Mixed FEM for the Hodge Laplacian 24.04.01. Virtual complexes can target the same Hodge-Laplacian and mixed PDE structures on more general meshes.

Historical & philosophical context Master

Virtual element methods emerged as a generalization of finite elements that can work on polygonal and polyhedral meshes while retaining computable projections and stability. The early VEM papers established the basic principle: the local space can be implicit if the numerical method only needs certain projected quantities ^{[Beirao da Veiga-Brezzi-Cangiani-Manzini-Marini-Russo]}.

The exterior-calculus direction adds FEEC compatibility to that idea. Rather than building one virtual space at a time, the compatible version builds a sequence of virtual spaces that mimic the de Rham complex ^{[Beirao da Veiga-Brezzi-Marini-Russo]}.

Philosophically, VEM exterior calculus separates what must be known from what need only exist. The basis functions may remain virtual, but the complex structure, projections, degrees of freedom, and stability estimates must be explicit enough to compute and prove convergence.

Bibliography Master

@article{BeiraoBrezziCangianiManziniMariniRusso2013VEM,
  author = {Beirao da Veiga, Lourenco and Brezzi, Franco and Cangiani, Andrea and Manzini, Gianmarco and Marini, L. Donatella and Russo, Alessandro},
  title = {Basic principles of virtual element methods},
  journal = {Mathematical Models and Methods in Applied Sciences},
  volume = {23},
  pages = {199--214},
  year = {2013}
}

@article{BeiraoBrezziMariniRusso2018VEMEC,
  author = {Beirao da Veiga, Lourenco and Brezzi, Franco and Marini, L. Donatella and Russo, Alessandro},
  title = {Virtual element implementation for general elliptic equations},
  journal = {Computer Methods in Applied Mechanics and Engineering},
  volume = {327},
  pages = {173--195},
  year = {2018}
}

@article{ArnoldFalkWinther2006VEMPointer,
  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}
}