Numerical Experiments for Darcy Flow on a Surface Using Mixed Exterior Calculus Methods

Hirani, Anil N.; Kalyanaraman, Kaushik

doi:10.48550/arxiv.1103.4865

Cited by 1 publication

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“…One main distinction from mimetic discretizations and support operator method is that those are designed for Euclidean domains and are dependent on the embedding. As a result, DEC can be used on simplicial approximations of non-flat manifolds as has been shown by others [52,59] and by the numerical experiments on a surface included in this paper and in [38]. There are methods such as those in [30,31,45,62] that are designed to work on specific surfaces.…”

Section: Introductionmentioning

confidence: 80%

“…The formulation is such that generalization to finite element exterior calculus involves the use of Whitney Hodge star instead of the primal-dual DEC Hodge star used in this paper. Experiments in this direction are an avenue for future work, some of which has been explored in [38]. Our method also generalizes in another way, to a dual formulation, and this has been done in [35].…”

Section: Discussionmentioning

confidence: 99%

“…However, there are no specific finite elements proposed and no experimental results shown. Recently, the first author of this paper and Kalyanaraman used the method proposed in this paper (and its generalization to finite element exterior calculus) for experiments on Darcy flow on a surface [38]. Since Darcy flow is equivalent to Poisson's equation in mixed form, the results in [38] are the first experimental results in this setting.…”

Section: Patch Tests In 2dmentioning

confidence: 96%

“…Recently, the first author of this paper and Kalyanaraman used the method proposed in this paper (and its generalization to finite element exterior calculus) for experiments on Darcy flow on a surface [38]. Since Darcy flow is equivalent to Poisson's equation in mixed form, the results in [38] are the first experimental results in this setting. We reproduce some of these experiments to show how easily our formulation generalizes to surfaces.…”

Section: Patch Tests In 2dmentioning

confidence: 96%

“…Indeed, by replacing the DEC Hodge star of our method by a Whitney Hodge star [8] one can obtain a finite element exterior calculus method for Darcy flow. This has been explored experimentally in [38]. As Bochev and Hyman point out in [11], all mimetic methods are related in that they all are cochain based methods.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Method for Darcy Flow Derived Using Discrete Exterior Calculus

Hirani

Nakshatrala

Chaudhry

2015

International Journal for Computational Methods in Engineering

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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.

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