Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes

Lipnikov, Konstantin; Morel, Jim E.; Shashkov, Mikhail

doi:10.1016/j.jcp.2004.02.016

Cited by 95 publications

(71 citation statements)

References 12 publications

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“…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 * . It is straightforward to extend the MFD method to locally refined meshes with hanging nodes [16], unstructured three-dimensional meshes composed of hexahedra, tetrahedra, and any cell type having three faces intersecting at each vertex.…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

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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“…Although we were at first surprised by the second order accurate numerical results, [52] and the references therein indicate the plausibility of this. In fact, it was [52] (and personal communications with the authors) that directly led to our first attempts to define unique pressure gradients on cell faces that could be used for both the large cell and all the small cells incident on a particular face.…”

Section: Second Order Accuracy On Octreesmentioning

confidence: 86%

“…In fact, it was [52] (and personal communications with the authors) that directly led to our first attempts to define unique pressure gradients on cell faces that could be used for both the large cell and all the small cells incident on a particular face.…”

Section: Second Order Accuracy On Octreesmentioning

confidence: 99%

Economic Analysis of National Nuclear Security Administration (NNSA) Modernization Alternatives

Hunter¹,

Bittle²,

Celec³

et al. 2007

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Since the seminal work of [92] on coupling the level set method of [69] to the equations for two-phase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes such as merging and pinching, as well as robust geometric information such as normals and curvature. Interestingly, this work also demonstrated the largest weakness of the level set method, i.e. mass or information loss characteristic of most Eulerian capturing techniques. In fact, [92] introduced a partial differential equation for battling this weakness, without which their work would not have been possible. In this paper, we discuss both historical and most recent works focused on improving the computational accuracy of the level set method focusing in part on applications related to incompressible flow due to both its popularity and stringent accuracy requirements. Thus, we discuss higher order accurate numerical methods such as Hamilton-Jacobi WENO [46], methods for maintaining a signed distance function, hybrid methods such as the particle level set method [27]

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“…Note that if one applies weak condition for the currents and strong conditions for the scalar fluxes, it results in interface relationships that are not valid for the scalar fluxes varying linearly along cell interfaces [49]. For example, in 2D, if the left face of i-th cell is a common face with the m-th and p-th cells, then we define strong continuity conditions for the normal components of the currents 30) and the weak continuity condition for the face-average scalar fluxes 31) where h iω is the cell-edge length.…”

Section: Interface Continuity Conditionsmentioning

confidence: 99%

Nonlinear Projective-Iteration Methods for Solving Transport Problems on Regular and Unstructured Grids

Anistratov¹,

Constantinescu²,

Roberts³

et al. 2007

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Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes

Cited by 95 publications

References 12 publications

Superconvergence of the Velocity in Mimetic Finite Difference Methods on Quadrilaterals

Superconvergence of the Velocity in Mimetic Finite Difference Methods on Quadrilaterals

Economic Analysis of National Nuclear Security Administration (NNSA) Modernization Alternatives

Nonlinear Projective-Iteration Methods for Solving Transport Problems on Regular and Unstructured Grids

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