A mixed finite volume method for elliptic problems

Mishev, Ilya D.; Chen, Qian-Yong

doi:10.1002/num.20213

Cited by 6 publications

(6 citation statements)

References 34 publications

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“…In contrast, primal-dual grid approaches, such as box integration [8], covolume [9,10], or finite volume [11] schemes use the two grids in the primal-dual complex to approximate the divergence and the gradient independently. When the two grids are topologically dual, such as in the case of Voronoi-Delauney triangulations or rectilinear 2 primal grids, these methods assume a particularly simple and elegant form.…”

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confidence: 99%

A High-Order Staggered Meshless Method for Elliptic Problems

Trask¹,

Perego²,

Bochev³

2017

SIAM J. Sci. Comput.

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Abstract. We present a new meshless method for scalar diffusion equations, which is motivated by their compatible discretizations on primal-dual grids. Unlike the latter though, our approach is truly meshless because it only requires the graph of nearby neighbor connectivity of the discretization points x i . This graph defines a local primal-dual grid complex with a virtual dual grid, in the sense that specification of the dual metric attributes is implicit in the method's construction. Our method combines a topological gradient operator on the local primal grid with a Generalized Moving Least Squares approximation of the divergence on the local dual grid. We show that the resulting approximation of the div-grad operator maintains polynomial reproduction to arbitrary orders and yields a meshless method, which attains O(h m ) convergence in both L 2 and H 1 norms, similar to mixed finite element methods. We demonstrate this convergence on curvilinear domains using manufactured solutions. Application of the new method to problems with discontinuous coefficients reveals solutions that are qualitatively similar to those of compatible mesh-based discretizations.

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confidence: 99%

A High-Order Staggered Meshless Method for Elliptic Problems

Trask¹,

Perego²,

Bochev³

2017

SIAM J. Sci. Comput.

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“…Usually the primary grid is used to approximate the scaler variable p, and the dual mesh is used to construct the discretization of the velocity. Different examples of primary and dual grids can be found in [2,16,24,25,26]. Consider the discrete problem:…”

Section: Mixed Finite Volume Methods Formulation For a Model Problemmentioning

confidence: 99%

“…Frequently the control volumes are formed by connecting the middle of the edges of each triangle with the center of mass of the triangle. The details of the method can be found in [26]. If the dual grid is Delauney mesh and the primary grid is Voronoi mesh, then we can approximate the scalar variable with piecewise constants, i.e., P h is the space of piecewise constants on the primary grid.…”

Section: More Mixed Finite Volume Methods and Their Multiscale Analoguesmentioning

confidence: 99%

“…Here we develop a framework for constructing mixed MsFV methods, which uses ideas from the mixed finite volume methods [24,25,26], multi-point flux approximations (MPFA) [2,16], and mixed MsFEM. The mixed MsFV are mass conservative methods, which is an important property of the discretizations used in subsurface flow simulations (see [11] for related discussion).…”

Section: Introductionmentioning

confidence: 99%

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Mixed Multiscale Finite Volume Methods for Elliptic Problems in Two-Phase Flow Simulations

Jiang

Mishev

2012

Commun. comput. phys.

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We develop a framework for constructing mixed multiscale finite volume methods for elliptic equations with multiple scales arising from flows in porous media. Some of the methods developed using the framework are already known [20]; others are new. New insight is gained for the known methods and extra flexibility is provided by the new methods. We give as an example a mixed MsFV on uniform mesh in 2-D. This method uses novel multiscale velocity basis functions that are suited for using global information, which is often needed to improve the accuracy of the multiscale simulations in the case of continuum scales with strong non-local features. The method efficiently captures the small effects on a coarse grid. We analyze the new mixed MsFV and apply it to solve two-phase flow equations in heterogeneous porous media. Numerical examples demonstrate the accuracy and efficiency of the proposed method for modeling the flows in porous media with non-separable and separable scales.

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“…G h is well defined as the FV approximation of the Poisson equation with homogeneous Dirichlet boundary conditions and an L 2 right hand side (see [9] and [12]). Furthermore, as follows from these papers (see also [8] and [18]), for any ψ ∈ L 2 (Ω) one has the error estimates…”

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confidence: 93%

Error estimates for the finite volume discretization for the porous medium equation

Pop

Sepúlveda²,

Radu

et al. 2010

Journal of Computational and Applied Mathematics

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A mixed finite volume method for elliptic problems

Cited by 6 publications

References 34 publications

A High-Order Staggered Meshless Method for Elliptic Problems

A High-Order Staggered Meshless Method for Elliptic Problems

Mixed Multiscale Finite Volume Methods for Elliptic Problems in Two-Phase Flow Simulations

Error estimates for the finite volume discretization for the porous medium equation

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