Tensorised Discrete de Rham Complex

• The topic defines tensorised discrete de Rham complexes as frameworks that integrate tensor-valued forms with classical de Rham operators, ensuring the cochain property and topological consistency.
• Methodologically, various constructions use discrete BGG diagrams, tensor-product spline spaces, and componentwise operations to decouple geometric discretizations from coefficient spaces.
• Analytical results demonstrate polynomial consistency, exact reconstruction, and effective static condensation on generic polygonal meshes, underpinning robust computational implementations.

Across recent work, the expression tensorised discrete de Rham complex denotes several closely related constructions rather than a single canonical object. In a polytopal Bernstein–Gelfand–Gelfand setting, it denotes the tensor-valued lower row of a discrete BGG diagram on generic polygonal meshes; in isogeometric analysis, it denotes the classical tensor-product spline de Rham complex on a parametric cube; and in tensor-valued exterior-calculus discretisations it denotes a de Rham complex tensored with a finite-dimensional coefficient space, so that differential operators act on the form degree while tensor indices are carried componentwise (Pietro et al., 23 Jul 2025, Patrizi et al., 2021, Oliynyk et al., 1 May 2025). What unifies these uses is preservation of the cochain property, compatibility with the continuous de Rham complex, and a discrete cohomology that reproduces the relevant continuous topological invariants.

1. Terminology and scope

The recent literature uses the term in at least three technically distinct senses.

Setting	Meaning of “tensorised”	Representative complex
Polygonal DDR–BGG construction	Vector/tensor-valued de Rham-type row, with symmetric restriction through a discrete skew kernel	$\underline X^{k+1}_{\GRAD,h}\to \underline X^{k+1}_{\VROT,h}\to \mathcal P^{k+1}(\mathcal T_h)^2$
Isogeometric toroidal discretisation	Tensor-product spline spaces on a parametric cube, pushed forward to the physical domain	$S^{p^r,p^s,p^t}\to X_1\to X_2\to X_3$
Tensor-valued exterior calculus	Product spaces such as $(X^k_{r,h})^3$, $(X^k_{r,h})^4$, or $\mathbb V\otimes X^k_{r,h}$	Componentwise discrete de Rham copies coupled algebraically

In the 2D polygonal work "Design and analysis of twisted and BGG Stokes-de Rham polytopal complexes", the bottom row of the discrete BGG diagram is explicitly described as a tensorised Discrete de Rham (DDR) complex of degree $k+1$, built from the serendipity DDR complex of Di Pietro–Droniou and coupled to a discrete Stokes complex (Pietro et al., 23 Jul 2025). In "Isogeometric de Rham complex discretization in solid toroidal domains", the phrase instead refers to the full tensor-product spline complex on the parametric domain $\Omega=[0,R]\times[0,S]\times[0,T]$, with grad, curl, and div represented by Kronecker products of one-dimensional difference matrices (Patrizi et al., 2021). In "A polytopal discrete de Rham scheme for the exterior calculus Einstein's equations", the tensorised aspect is that tensor fields are represented as several copies of the discrete de Rham complex, effectively $\mathbb V\otimes X^k_{r,h}$, with the discrete exterior derivative acting componentwise (Oliynyk et al., 1 May 2025).

A recurrent misconception is to identify tensorised only with tensor-product cells or tensor-product basis functions. The polygonal BGG construction shows a different usage: there the complex lives on generic polygonal meshes and the tensorisation is tied to vector/tensor-valued discrete forms and to the symmetric-tensor restriction through a discrete skew operator (Pietro et al., 23 Jul 2025). This suggests that the phrase is best understood structurally, not purely geometrically.

2. Polytopal DDR construction on generic polygonal meshes

In the polygonal BGG framework, the tensorised DDR complex is the lower row of a fully discrete anti-commuting diagram in 2D: $0 \longrightarrow \underline X^{k+1}_{\GRAD,h} \xrightarrow{\ \nabla_h^{k+1}\ } \underline X^{k+1}_{\VROT,h} \xrightarrow{\ \ROT_h^{k+1}\ } \mathcal P^{k+1}(\mathcal T_h)^2 \longrightarrow 0.$ The mesh is polytopal, with polygonal elements $T$, edges $S^{p^r,p^s,p^t}\to X_1\to X_2\to X_3$0, and vertices $S^{p^r,p^s,p^t}\to X_1\to X_2\to X_3$1. The gradient-type space is

$S^{p^r,p^s,p^t}\to X_1\to X_2\to X_3$2

while the rotor-type space is

$S^{p^r,p^s,p^t}\to X_1\to X_2\to X_3$3

The target is the broken polynomial vector space $S^{p^r,p^s,p^t}\to X_1\to X_2\to X_3$4 (Pietro et al., 23 Jul 2025).

The local discrete operators are defined by integration-by-parts identities. On each edge, the discrete edge gradient $S^{p^r,p^s,p^t}\to X_1\to X_2\to X_3$5 is reconstructed from edge and vertex DOFs. On each cell, the discrete cell gradient $S^{p^r,p^s,p^t}\to X_1\to X_2\to X_3$6 is defined against test tensors through a discrete divergence formula. The global gradient is then

$S^{p^r,p^s,p^t}\to X_1\to X_2\to X_3$7

Dually, the local discrete rotor $S^{p^r,p^s,p^t}\to X_1\to X_2\to X_3$8 is defined by

$S^{p^r,p^s,p^t}\to X_1\to X_2\to X_3$9

and the global discrete rotor is $(X^k_{r,h})^3$0 (Pietro et al., 23 Jul 2025).

The tensorised character is explicitly twofold. First, the spaces are already vector/tensor-valued, matching the continuous lower de Rham row $(X^k_{r,h})^3$1. Second, the BGG output uses the symmetric subspace

$(X^k_{r,h})^3$2

so that the cell tensors satisfy $(X^k_{r,h})^3$3. The paper emphasizes that the construction is serendipity DDR: some cell polynomial components are reduced, while maintaining the same polynomial approximation order for the discrete operators, and avoiding trimmed spaces (Pietro et al., 23 Jul 2025). More abstractly, serendipity discrete complexes with enhanced regularity are generated from extension and reduction maps, with preserved cohomology, in the framework of "Serendipity discrete complexes with enhanced regularity" (Pietro et al., 2024).

3. Exactness, cohomology, and homological structure

The tensorised DDR row is a genuine complex: $(X^k_{r,h})^3$4 In the discrete BGG outputs, the cohomology is shown to be isomorphic to the corresponding continuous cohomologies and hence to sums of de Rham cohomologies. For the discrete Hessian complex

$(X^k_{r,h})^3$5

the kernel of the discrete Hessian is exactly the discrete affine functions,

$(X^k_{r,h})^3$6

and the discrete rotor

$(X^k_{r,h})^3$7

is surjective. A dimension-count argument then shows that the kernel, image, and cohomology of the discrete Hessian complex have the same dimensions as in the continuous Hessian complex, for arbitrary topology (Pietro et al., 23 Jul 2025).

This homological behavior is consistent with a broader line of work in which discrete complexes are built to be cohomologically equivalent to the continuous de Rham complex. The abstract construction in (Pietro et al., 2024) produces a fourth complex whose cohomology is isomorphic to that of three given complexes linked by extension and reduction maps. In the fully discrete polytopal exterior-calculus framework, the DDR and VEM complexes are likewise shown to have cohomology isomorphic to the cohomology of the continuous de Rham complex (Bonaldi et al., 2023). A closely related manifold version proves the same statement for polytopal complexes on curved manifolds (Droniou et al., 2024). Together, these results make clear that tensorisation is compatible with homological correctness, provided the cochain maps and local polynomial structures are designed appropriately.

4. Anti-commuting BGG diagrams and tensorial complexes

The polygonal tensorised DDR complex is not an isolated de Rham row; it is the lower row of an anti-commuting discrete BGG diagram whose upper row is a discrete Stokes complex. The decisive algebraic identity is

$(X^k_{r,h})^3$8

This anti-commutation produces two derived complexes. The first is a twisted complex, which discretises Reissner–Mindlin or Cosserat-type models. The second is the BGG Hessian/elasticity complex obtained by restricting to the kernel and cokernel of $(X^k_{r,h})^3$9, namely the symmetric-stress complex $(X^k_{r,h})^4$0. In this way, the tensorised DDR row provides the discrete $(X^k_{r,h})^4$1-like tensor space, the symmetric stress space, and the discrete rotor needed for Kirchhoff–Love plates and, in rotated form, Hellinger–Reissner elasticity on generic polygonal meshes (Pietro et al., 23 Jul 2025).

This BGG interpretation belongs to a broader tensorial de Rham tradition. In "Differential Complexes in Continuum Mechanics", tensorial grad–curl–div complexes are identified with vector-valued de Rham complexes, so that their cohomology is essentially tensorised de Rham cohomology (Angoshtari et al., 2013). In "Distributional Hessian and divdiv complexes on triangulation and cohomology", discrete Hessian and divdiv complexes on triangulations are constructed as discrete BGG complexes with local polynomial shape functions and various types of Dirac measure on subsimplices, and their cohomology is shown to be isomorphic to de Rham cohomology with coefficients (Hu et al., 2023). A plausible implication is that polygonal tensorised DDR complexes and triangulation-based distributional BGG complexes are different discrete realisations of the same BGG program: de Rham structure first, tensorial mechanics second.

5. Analytical properties and implementation

The analytical backbone of the tensorised DDR complex is polynomial consistency, reconstruction stability, and transferred Poincaré control. In the polygonal BGG construction, the discrete gradient and rotor are exact on polynomials up to degree $(X^k_{r,h})^4$2: $(X^k_{r,h})^4$3 for the stated polynomial classes. For the discrete Hessian, the local reconstruction satisfies

$(X^k_{r,h})^4$4

which is the degree $(X^k_{r,h})^4$5 consistency of the discrete Hessian. The framework also uses potential reconstructions $(X^k_{r,h})^4$6, stabilised discrete scalar products, and norm equivalences, and it transfers Poincaré inequalities from the lowest-order DDR(0) complex to richer complexes through cochain maps (Pietro et al., 23 Jul 2025).

Implementation follows the standard DDR pattern but in a tensor-valued setting. The construction is fully polytopal; elements are arbitrary polygons, possibly with hanging nodes. The DOFs are cell polynomials, edge polynomials, and vertex values, without sub-triangulation of each polygon. All reconstructors and discrete operators are local and are defined elementwise by integration-by-parts relations. This locality makes static condensation possible: cell DOFs can be eliminated elementwise in many discretisations, leaving a global system in terms of edge and vertex unknowns. At the element level, all spaces are composed of full polynomial spaces—no trimmed polynomial subspaces—which the paper identifies as a major implementation advantage (Pietro et al., 23 Jul 2025).

These analytical themes recur throughout the wider discrete de Rham literature. "Local Bounded Commuting Projection Operator for Discrete de Rham Complexes" constructs local bounded commuting projection operators for nonstandard finite element de Rham complexes in two and three dimensions (Hu et al., 2023). "An exterior calculus framework for polytopal methods" establishes commutation properties between interpolators and discrete and continuous exterior derivatives, proves key polynomial consistency results, and shows that the resulting cohomologies are isomorphic to the cohomology of the continuous de Rham complex (Bonaldi et al., 2023). The tensorised DDR complex fits naturally into this pattern: its additional tensor structure does not replace the usual discrete de Rham requirements, but intensifies them.

6. Tensor-product, manifold, and DEC variants

A genuinely tensor-product realisation appears in "Isogeometric de Rham complex discretization in solid toroidal domains". There the full tensorised discrete de Rham complex on the parametric cube

$(X^k_{r,h})^4$7

is the classical IgA spline complex

$(X^k_{r,h})^4$8

with grad, curl, and div represented by Kronecker products of one-dimensional difference matrices. Because the toroidal geometry map is singular on the polar curve, the full tensor-product spaces must be restricted by extraction operators $(X^k_{r,h})^4$9. The resulting reduced complex preserves the torus cohomology dimensions

$\mathbb V\otimes X^k_{r,h}$0

matching the Betti numbers of the solid torus (Patrizi et al., 2021).

A manifold extension is provided by "A polytopal discrete de Rham complex on manifolds, with application to the Maxwell equations". There the local polynomial spaces are built by pullback through cellwise charts $\mathbb V\otimes X^k_{r,h}$1, with compatibility assumptions ensuring that traces of trimmed polynomials remain trimmed polynomials on lower-dimensional entities. The resulting discrete complex has the same cohomology as the continuous de Rham complex, arbitrary order of accuracy, and can in principle be designed on meshes made of generic curved elements. In the Maxwell application on a 2D manifold without boundary, the discrete Gauss constraint is preserved because the complex property $\mathbb V\otimes X^k_{r,h}$2 is built into the scheme (Droniou et al., 2024).

At the chain level, tensorisation is especially explicit in discrete exterior calculus. In "2D discrete Hodge-Dirac operator on the torus", the combinatorial plane is built as the tensor product chain complex $\mathbb V\otimes X^k_{r,h}$3, the coboundary $\mathbb V\otimes X^k_{r,h}$4 gives a discrete de Rham complex, and the periodic $\mathbb V\otimes X^k_{r,h}$5 block yields a combinatorial torus with discrete Hodge decomposition and cohomology

$\mathbb V\otimes X^k_{r,h}$6

for the $\mathbb V\otimes X^k_{r,h}$7 torus model (Sushch, 2022). In a different direction, the exterior-calculus discretisation of Einstein’s equations uses several coupled copies of a polytopal discrete de Rham complex, effectively $\mathbb V\otimes X^k_{r,h}$8, to represent tensor-valued differential forms while letting the discrete exterior derivative act componentwise (Oliynyk et al., 1 May 2025). These variants make clear that tensorisation can enter through product geometry, through coefficient spaces, through BGG algebra, or through all three at once.