Papers › Variational formulations of transport on combinatorial meshes
Variational formulations of transport on combinatorial meshes
Kiprian Berbatov, Andrey P. Jivkov
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
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.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections