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We derive a numerical method for Darcy flow, hence also for Poisson'sequation in mixed (first order) form, based on discrete exterior calculus(DEC). Exterior calculus is a generalization of vector calculus to smoothmanifolds and DEC is one of its discretizations on simplicial complexes such astriangle and tetrahedral meshes. DEC is a coordinate invariant discretization,in that it does not depend on the embedding of the simplices or the whole mesh.We start by rewriting the governing equations of Darcy flow using the languageof exterior calculus. This yields a formulation in terms of flux differentialform and pressure. The numerical method is then derived by using the frameworkprovided by DEC for discretizing differential forms and operators that act onforms. We also develop a discretization for spatially dependent Hodge star thatvaries with the permeability of the medium. This also allows us to addressdiscontinuous permeability. The matrix representation for our discretenon-homogeneous Hodge star is diagonal, with positive diagonal entries. Theresulting linear system of equations for flux and pressure are saddle type,with a diagonal matrix as the top left block. The performance of the proposednumerical method is illustrated on many standard test problems. These includepatch tests in two and three dimensions, comparison with analytically knownsolution in two dimensions, layered medium with alternating permeabilityvalues, and a test with a change in permeability along the flow direction. Wealso show numerical evidence of convergence of the flux and the pressure. Aconvergence experiment is also included for Darcy flow on a surface. A shortintroduction to the relevant parts of smooth and discrete exterior calculus isincluded in this paper. We also include a discussion of the boundary conditionin terms of exterior calculus.

Numerical method for Darcy flow derived using Discrete Exterior Calculus