Finite element systems of differential forms

1 Introduction

• Mixed finite elements for grad, curl, and div provide compatible trial and test spaces, while differential-form sequences organize these operators as instances of the exterior derivative.Prior high-order constructions include simplex elements and tensor-product spaces on Cartesian products.Source: §1 · pp. 2–3
• Natural degrees of freedom may not be bounded on the relevant Banach spaces, while eigenvalue convergence requires stronger interpolation bounds such as L2 stability.This lack of continuity also hampers high-order approximation.Source: §1 · p. 2
• The framework targets polyhedral meshes and non-polynomial functions, including piecewise functions motivated by dual meshes and upwinding.The dual of a simplicial mesh is not simplicial, motivating finite element spaces on general polytopes.Source: §1 · p. 3
• Cellwise restriction surjectivity and exactness under the exterior derivative define compatibility, yielding commuting interpolators and global topological properties.Degrees of freedom are deduced from the system rather than built into its definition.Source: §1 · p. 3
• The paper combines interpolators with smoothers and extensions to obtain Lq-stable commuting projections, with locally polynomial-preserving constructions and new Sobolev injection and translation estimates.The estimates extend the analysis from L2 to all finite q.Source: §1 · p. 4

• A cellular complex decomposes a compact metric space into finitely many cells with disjoint interiors and cellwise boundaries; intersections of two cells are unions of cells.Source: §2 · pp. 4–5
• Cellwise integration defines the De Rham map to incidence cochains, and its diagram with exterior differentiation and coboundary commutes by Stokes’ theorem.Source: §2 · p. 6
• A finite element system assigns differential-form spaces to cells, stable under exterior differentiation and pullback along cell inclusions; global spaces are inverse limits enforcing matching pullbacks.Piecewise forms match across interfaces in the sense of equal pullbacks.Source: §2 · pp. 7–9
• On flat cell complexes, polynomial form orders satisfying the stated nesting and degree conditions produce finite element systems; trimmed polynomial systems are a named example.Source: §2 · p. 8
• For a compatible element system, the De Rham map induces isomorphisms in cohomology.Compatibility means the system admits extensions and is locally exact.Source: §2 · p. 9
• If extensions hold and point cells carry constants, top-degree spaces contain a form with nonzero integral, and global k-form dimensions are at least the number of k-cells.Source: §2 · pp. 9–10

Finite element systems

• Exactness of the element-system sequence is equivalent to each A0(T) containing constants and the boundary-condition sequence being exact on each cell.The boundary-condition sequence runs from A0_0(T) through A^dim T(T) to R.Source: §2 · p. 10

Tensor products

• Tensor products have degree-zero spaces equal to the tensor products of the factors’ degree-zero spaces, and preserve extensions and local exactness when both factors have those properties.The extension and local-exactness results apply when both factor systems admit extensions or are locally exact, respectively.Source: §2 · pp. 11–12

Nesting

• If each parameter-indexed system is compatible, the nested system defined by an order-preserving parameter function is also compatible.The construction restricts each cell’s space to elements whose restrictions belong to the spaces selected on its subcells; it includes variable-order trimmed systems used for hp-methods.Source: §2 · p. 13

Locally harmonic forms

• Under sequence (33)’s exactness, each prescribed top-degree integral has a unique A-harmonic representative, and extendible lower-degree boundary data have unique A-harmonic extensions.These uniqueness results assume a scalar product on the element spaces.Source: §2 · p. 13
• For a compatible system, the subsystem of locally harmonic forms is compatible and its De Rham map to cellular cochains is an isomorphism.The canonical basis induced by the cellular-cochain basis can be constructed by recursive harmonic extension.Source: §2 · p. 14
• Locally harmonic spaces support dual-mesh finite elements with primal-dual form-space dimension matching, and recursive harmonic subspaces on agglomerated meshes provide a multilevel construction for multigrid preconditioning.The dual-mesh construction uses a simplicial refinement and has a proved inf-sup duality; agglomeration retains the fine-mesh function spaces.Source: §2 · p. 14
• For convection problems, standard-L2 trial and weighted-L2 test harmonic spaces yield canonical bases that are upwinded or downwinded relative to one another.On each cell, the construction uses an affine function βT with dβT equal to the chosen constant approximation of the covariant field; exp(βT)v satisfies the stated equations for constant v.Source: §2 · pp. 15–16
• For Helmholtz approximation, locally harmonic forms can contain a plane wave away from interior cell resonances, while the bilinear form degenerates at those resonances.More generally, the construction can approximate solutions locally represented by plane waves using cellwise functions βT, potentially chosen adaptively.Source: §2 · p. 16

Mirror systems and interpolators

• A mirror system is faithful exactly when its duality pairing with the element system’s interior degrees of freedom is invertible, and faithful mirrors yield interpolators.Faithful mirror systems, interpolators, and extensions are in bijection when the element system admits extensions.Source: §3 · pp. 17–18

Commuting interpolators

• An interpolator commutes with the exterior derivative precisely when its mirror system satisfies the stated compatibility condition.The canonical mirror system for trimmed polynomials of order p gives a commuting interpolator.Source: §3 · p. 19

Examples and constructions

• Tensor products of faithful mirror systems remain faithful, and trimmed polynomial finite element systems of order p are compatible.The trimmed polynomial system is also described as minimal among compatible finite element systems containing polynomial differential forms of order p − 1.Source: §3 · pp. 18–19

Commuting extensions

• For compatible finite element systems, harmonic extension operators commute when the cell sequence is exact and extensions exist.The proposition applies to scalar-product-equipped elements on a cell.Source: §3 · p. 20

Extension-projection interpolators

• When projections commute with the exterior derivative and preserve integrals, and extensions commute, the associated extension-projection interpolator has both properties.The construction proceeds inductively over cell dimension.Source: §3 · pp. 20–21

Approximation properties

• Defining an interpolator from extensions and projections lets its approximation properties follow directly from estimates on those operators.This is presented as an advantage over constructing interpolators through mirror systems.Source: §3 · p. 21

4 Quasi-interpolators

• Stable quasi-interpolators combine an interpolator, a regularizer, and an extension operator; commuting interpolators are order-optimal when the finite elements contain polynomials of degree ℓ−1.The h-setting assumes compact families of pulled-back finite elements, normally requiring shape-regular cells and excluding unbounded polynomial degree.Source: §4 · pp. 22–23
• The regularizer associated with Φ commutes with the exterior derivative when Φ has continuous second derivatives and its parameter map is a diffeomorphism.It is constructed using pullbacks, with a smooth, rotationally invariant, nonnegative kernel supported in the unit ball and having integral 1.Source: §4 · p. 23
• The regularizer maps L1 forms to continuously differentiable forms with controlled operator norm, and also admits estimates with equal differentiation orders on both sides.The section states these smoothing and derivative-estimate results under bounds on the deformation maps Φ.Source: §4 · p. 25
• For polynomial forms of degree at most p, the regularizer reproduces every derivative at x when the form is polynomial on a neighborhood of x.The kernel is chosen to preserve polynomials up to the required degree.Source: §4 · p. 26
• The regularizer is adapted to the mesh through cellwise scaling, with estimates stated for individual cells, top-dimensional cells, and forms on a neighborhood of the domain.The section defines a macroelement as the union of cells touching a given cell and introduces estimates on a small neighborhood of S.Source: §4 · pp. 27–28
• Extension operators are designed to preserve polynomials, commute with the exterior derivative, and be continuous in Wℓ,q norms for ℓ≤p+1.They use pullbacks by maps from outside S into S, combined linearly to obtain continuity and polynomial preservation.Source: §4 · p. 28

Extension operator

• The extension construction meets its stated conditions for Lipschitz boundaries by regularizing a vector-field-based construction whose initial form lacks regularity.The construction uses a regularized distance function.Source: §4 · p. 31
• The extension operator E preserves Cℓ regularity and maps polynomials of degree at most p to polynomials of degree at most p on neighborhoods of touching cells.Source: §4 · p. 33

Quasi-interpolators

• Theorem 4.15 establishes quasi-interpolators Qε that commute with the exterior derivative when it is in Lq(S).The theorem also specifies a choice of ε in relation to ε′.Source: §4 · p. 34

Projections

• The operators Pn are Lq(S)-stable projections onto Xk_n that commute with the exterior derivative.Source: §4 · p. 34

Eigenvalue convergence

• For q = 2, the Galerkin discretization of d⋆d on Xk_n yields eigenvalue convergence for the Hodge-Laplacian in mixed form.Source: §4 · p. 34

• The Sobolev injection theorem generalizes the authors’ earlier result to differential forms in all dimensions.Source: §5 · p. 34
• The translation estimate introduced in has an optimal bound and generalizes to all known mixed finite elements in the h-version.Source: §5 · p. 34
• The broken H1 seminorm includes facet-jump terms scaled to match the scaling of the other summed terms.Source: §5 · pp. 34–35
• The analysis uses spaces X_n^k=A[n]_k(T_n) from compatible finite element systems on a sequence of cellular complexes.Source: §5 · p. 35

• If f is short, then g(x)=x+f(x) is a Lipschitz bijection onto an open range with a Lipschitz inverse.The inverse Lipschitz constant is no worse than (1−δ)^−1.Source: Appendix · p. 37
• If the outward normal has a positive component along a unit vector m near a boundary point, the boundary is locally a Lipschitz graph over the plane orthogonal to m.Source: Appendix · p. 38
• For a bounded domain with locally Lipschitz boundary, a smooth unit outward vector field gives a collar map that is a Lipschitz isomorphism onto its range.Source: Appendix · p. 39