Variational formulations of transport phenomena on combinatorial meshes

We develop primal and mixed variational formulations of transport phenomena on cell complexes with simple polytope connectivity. This framework addresses materials with internal structures comprising components of different topological dimensions, where cells of each dimension may possess distinct physical properties. The approach, which we call Combinatorial Mesh Calculus (CMC), extends Forman’s combinatorial differential forms, previously used to formulate strong conservation laws. CMC operates directly on meshes without requiring smooth embeddings, using discrete analogues of the exterior derivative, Hodge star, and co-differential operators. Our mixed formulation leads to a block-diagonal mass-like matrix arising from inner products weighted by material coefficients, enabling efficient local elimination strategies within the mixed system. CMC differs from Discrete Exterior Calculus, which requires circumcentric duality and well-centred meshes, and from Finite Element Exterior Calculus, which constructs polynomial spaces on smooth domains. Our framework applies to general cell complexes, including curved cells and irregular meshes; nonetheless irregularity leads to worse numerical performance. The mathematical development proceeds in parallel between the smooth and discrete settings, establishing correspondences between continuous and discrete operators. Initial boundary value problems are formulated for mass diffusion, heat conduction, charge transport, and fluid flow through porous media. Numerical examples on regular and irregular meshes in two and three dimensions demonstrate agreement with analytical solutions. The framework enables modelling of transport in materials where microstructural topology influences macroscopic behaviour, with applications to polycrystalline materials, composites, and porous media.

💡 Research Summary

The paper introduces a novel discrete framework called Combinatorial Mesh Calculus (CMC) for modelling transport phenomena (mass diffusion, heat conduction, charge transport, and fluid flow in porous media) on cell complexes that may contain components of different topological dimensions. The authors start by motivating the need for a representation that can capture the multi‑dimensional microstructure of modern materials—grains, grain boundaries, and junction lines—each with distinct material properties, something that traditional continuum discretisations (FEM, FDM, FVM) struggle to handle without artificial smoothing or complex enrichment techniques.

Section 2 provides a concise review of exterior calculus on smooth manifolds, emphasizing the distinction between the purely topological exterior derivative (d) and the metric‑dependent Hodge star (*) and codifferential (\delta). This language is chosen because differential forms naturally associate physical quantities with the geometric domains they live on (0‑forms on vertices, (D‑1)‑forms on faces, etc.), and because the same operators admit clean discrete analogues.

In Section 3 the continuous transport equations are rewritten using exterior calculus. Two variational formulations are derived: a primal weak form where the potential is a 0‑form, and a mixed weak form where the flux is a ((D-1))‑form and the dual potential a (D)‑form. Both formulations are expressed entirely in terms of (d), (\delta), the Hodge inner product, and material coefficients. The mixed formulation is shown to be a generalisation of the classic Dirichlet‑Neumann mixed problem and closely related to the mixed formulations appearing in Finite Element Exterior Calculus (FEEC).

Section 4 builds the discrete side. The authors define a “combinatorial mesh” as an abstract cell complex equipped with a Forman subdivision, allowing non‑convex, curved, or irregular cells. They extend Forman’s combinatorial differential forms (cochains) with a metric‑dependent inner product and a Hodge star that respects the material coefficients. Importantly, convexity is no longer required, which distinguishes CMC from Discrete Exterior Calculus (DEC) that relies on circumcentric duals and well‑centred meshes.

Section 5 translates the continuous variational statements into discrete ones by replacing (d,\delta, ) with their discrete counterparts (d_h,\delta_h,_h) and the continuous inner product with a weighted discrete inner product (\langle\cdot,\cdot\rangle_h). The resulting discrete mixed system possesses a block‑diagonal “mass‑like” matrix (M_h) that is assembled locally per cell dimension. Because of this structure, the flux variables can be eliminated element‑wise, leaving a reduced system for the potential alone. This local elimination is computationally cheap and avoids the global saddle‑point problems typical of mixed FEM.

Section 6 presents numerical experiments. The authors test steady‑state diffusion, heat, electrostatic, and Darcy flow problems on regular square/hexagonal meshes in 2 D, regular hexahedral meshes in 3 D, and on highly irregular Delaunay‑type meshes. Manufactured solutions are used to assess convergence. All cases exhibit first‑order convergence in the (L^2) norm, with regular meshes achieving rates close to the theoretical optimum. Irregular meshes show larger error constants due to distorted cell geometry, but the convergence trend remains robust. The block‑diagonal mass matrix enables efficient static condensation, dramatically reducing the size of the global linear system.

The authors conclude that CMC provides a mathematically rigorous, dimension‑agnostic way to model transport on complex microstructures without imposing geometric constraints on the mesh. It bridges continuous exterior calculus and discrete combinatorial topology, yields efficient linear algebraic structures, and is readily extensible to nonlinear, time‑dependent, and multiphysics problems. Future work is suggested on optimal weighting for highly irregular meshes, adaptive mesh refinement within the CMC framework, and high‑performance parallel implementations.