Numerical Method for Darcy Flow Derived Using Discrete Exterior Calculus

Abstract

Abstract. We derive a numerical method for Darcy flow, hence also for Poisson's equation in first order form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is its discretization on simplicial complexes such as triangle and tetrahedral meshes. We start by rewriting the governing equations of Darcy flow using the language of exterior calculus. This yields a formulation in terms of flux differential form and pressure. The numerical method is then derived by using the framework provided by DEC for discretizing differential forms and operators that act on forms. We also develop a discretization for spatially dependent Hodge star that varies with the permeability of the medium. This also allows us to address discontinuous permeability. The matrix representation for our discrete non-homogeneous Hodge star is diagonal, with positive diagonal entries. The resulting linear system of equations for flux and pressure are saddle type, with a diagonal matrix as the top left block. Our method requires the use of meshes in which each simplex contains its circumcenter. The performance of the proposed numerical method is illustrated on many standard test problems. These include patch tests in two and three dimensions, comparison with analytically known solution in two dimensions, layered medium with alternating permeability values, and a test with a change in permeability along the flow direction. A short introduction to the relevant parts of smooth and discrete exterior calculus is included in this paper. We also include a discussion of the boundary condition in terms of exterior calculus.

Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model

“…Here α 0 , α 1 , α 2 and K 0 , K 1 , K 2 are the conductivities and permeabilities at the three hierarchical levels and are given in equation (24) and (26). The homogeneous Neumann boundary conditions (2a) complete the system.…”

“…One of the authors recently showed that a formulation based on exterior calculus is very effective in simulating Darcy flow on curved surfaces [26]. However, the computational approach in the present article differs from that discussed in [26] since here we consider: (i) an extension of the Darcy equations to the flow through a hierarchical porous medium, thereby requiring suitable discretization of the corresponding hierarchical variable θ; (ii) source terms distributed along the centerlines of the vascular trees, thereby requiring suitable discretization of the corresponding delta functions.…”

“…However, the computational approach in the present article differs from that discussed in [26] since here we consider: (i) an extension of the Darcy equations to the flow through a hierarchical porous medium, thereby requiring suitable discretization of the corresponding hierarchical variable θ; (ii) source terms distributed along the centerlines of the vascular trees, thereby requiring suitable discretization of the corresponding delta functions. In addition, we remark that the method in [26] relies on a mixed formulation of the problem, whereas here we solve for the pressure as the sole primal variable.…”

A review of some geometric integrators

“…More recent works in electromagnetism include [106][107][108][109]. In the field of mechanics, DEC has been employed to solve Darcy, Euler and Navier-Stokes equations in some basic configurations [110][111][112], to geometrizes elasticity problems [113,114] or to solve port-Hamiltonian systems [115]. DEC belongs to the family of cochain-based mimetic discretization methods described in [116].…”

“…We present here briefly a DEC theory which follows the approach of Hirani and his coauthors [111,112,214]. More details can be found in the literature on algebraic topology [215][216][217][218][219] and on DEC [89][90][91][92][99][100][101][102][103]214,[220][221][222].…”