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.