The Numerical Solution of Diffusion Problems in Strongly Heterogeneous Non-isotropic Materials

Hyman, James M.; Shashkov, Mikhail; Steinberg, Stanly

doi:10.1006/jcph.1996.5633

Cited by 181 publications

(152 citation statements)

References 15 publications

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“…We refer the reader to the overview compiled by Morvan [Mor01]. Note also the tight connection with the "Mimetic Discretizations" used in computational physics by Shashkov, Hyman, and Steinberg [HS97,HSS97]. Although it shares a lot of similarities with all these approaches, our work offers a different, unified derivation that ensures accuracy and tight error bounds, leading to simple formulae that are straightforward to implement.…”

Section: Previous Workmentioning

confidence: 95%

Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

Meyer

Desbrun

Schröder

et al. 2003

Mathematics and Visualization

1,478

1,168

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Summary. This paper proposes a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures on arbitrary triangle meshes. We present a consistent derivation of these first and second order differential properties using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method, and compare them to existing formulations. Building upon previous work in discrete geometry, these operators are closely related to the continuous case, guaranteeing an appropriate extension from the continuous to the discrete setting: they respect most intrinsic properties of the continuous differential operators. We show that these estimates are optimal in accuracy under mild smoothness conditions, and demonstrate their numerical quality. We also present applications of these operators, such as mesh smoothing, enhancement, and quality checking, and show results of denoising in higher dimensions, such as for tensor images.

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Section: Previous Workmentioning

confidence: 95%

Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

Meyer

Desbrun

Schröder

et al. 2003

Mathematics and Visualization

1,478

1,168

View full text Add to dashboard Cite

show abstract

“…The use of the geometric center instead of the mass center is due to the following property of the MFD method. The method is exact for linear solutions when the pressure variable, p(c i ), is evaluated at the geometric center c i [15]. The second-order convergence rate is observed for both the pressure and velocity variables in the discrete L 2 -and L ∞ -norms.…”

Section: Numerical Experimentsmentioning

confidence: 97%

“…Conservation laws, solution symmetries, and the fundamental identities and theorems of vector and tensor calculus are examples of such properties. This "mimetic" technique has been applied successfully to several applications including diffusion [22,15,18], magnetic diffusion and electromagnetics [14], continuum mechanics [17], and gas dynamics [8]. For problem (1.1), the mimetic technique uses discrete flux G and divergence DIV operators for the continuum operators −Kgrad and div, respectively, which are adjoint to each other, i.e., G = DIV * .…”

Section: Introductionmentioning

confidence: 99%

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Superconvergence of the Velocity in Mimetic Finite Difference Methods on Quadrilaterals

Berndt¹,

Lipnikov²,

Shashkov³

et al. 2005

SIAM J. Numer. Anal.

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Abstract. Superconvergence of the velocity is established for mimetic finite difference approximations of second-order elliptic problems over h 2 -uniform quadrilateral meshes. The superconvergence result holds for a full tensor coefficient. The analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space. The theoretical results are confirmed by numerical experiments.Key words. mixed finite element, mimetic finite difference, tensor coefficient, superconvergence AMS subject classifications. 65N06, 65N12, 65N15, 65N22, 65N30 DOI. 10.1137/040606831 1. Introduction. We consider the numerical approximation of a linear secondorder elliptic problem. In porous medium applications, this equation models single phase Darcy flow and is usually written as a first-order system for the fluid pressure p and velocity u:

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“…The mimetic finite difference (MFD) method has been successfully employed for solving problems of continuum mechanics [19], electromagnetics [13], gas dynamics [7], and linear diffusion on polygonal and polyhedral meshes in both the Cartesian and polar coordinates [14,20,18]. The MFD method mimics essential properties of the continuum equations, such as conservation laws, solution symmetries, and the fundamental identities and theorems of vector and tensor calculus.…”

Section: Introductionmentioning

confidence: 99%

Local flux mimetic finite difference methods

2008

Self Cite

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We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral grids. To approximate the flux (vector variable), the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per (n-gon) element face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom per element. A specially chosen inner product in the space of discrete fluxes allows for local flux elimination and reduction of the method to a symmetric cell-centered finite difference scheme for the pressure. In the case of simplicial grids, optimal first-order convergence is proved for both variables, as well as second-order convergence for the scalar variable. Numerical results confirm the theory.

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The Numerical Solution of Diffusion Problems in Strongly Heterogeneous Non-isotropic Materials

Cited by 181 publications

References 15 publications

Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

Superconvergence of the Velocity in Mimetic Finite Difference Methods on Quadrilaterals

Local flux mimetic finite difference methods

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