Variational formulations of transport on combinatorial meshes

We develop analogues of the primal and mixed variational formulations of the conservation laws describing transport of scalar physical properties for topological spaces referred to as cell complexes. The development is restricted to cell complexes with cells having connectivity of simple polytopes. Such spaces are suitable representations of physical systems composed of elements of different apparent topological dimensions, where all elements might have individual properties, and elements of a given dimension might interact via common elements of lower dimensions. This modelling foundation can be considered as intermediate between the foundation for particle-based modelling, which is the discrete topology, and the foundation for continuum modelling, which is the smooth topology. The new foundation offers advantages for the analysis of the behaviour of physical systems with complex internal structures. The development is based on a calculus with combinatorial differential forms which is a discrete analogue of the smooth exterior calculus with differential forms. We call the resulting calculus combinatorial mesh calculus (CMC) as it is based on combinatorial meshes, for which embedding is forgotten and only the connectivity between cells and measures of the cells are used in calculations. We discuss how the obtained formulation is specialised for several different problems, including mass diffusion, heat conduction, fluid flow in porous media, and charge diffusion, and provide details about the formulation of initial boundary value problems for these transport cases. Examples and results for selected boundary value problems are given to demonstrate the capabilities of the CMC formulations.