Analysis and optimization of inner products for mimetic finite difference methods on a triangular grid

Liska, R.; Shashkov, Mikhail; Ganzha, Victor

doi:10.1016/j.matcom.2004.05.008

Cited by 17 publications

(10 citation statements)

References 6 publications

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“…The global resulting system therefore involves only the velocity unknowns at the faces of the mesh. The HMM methods contain the Mimetic Finite Difference schemes which, for triangular meshes and particular choices of β (i) K , are algebraically identical to the lowest order Raviart-Thomas method [26]. The hybridisation of this method gives the same matrix on the edge unknowns as the Crouzeix-Raviart scheme [6].…”

Section: The Linear Mapping DIVmentioning

confidence: 99%

Gradient Schemes for Stokes problem

Droniou¹,

Eymard

Feron

2015

IMA J Numer Anal

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The gradient scheme framework encompasses several conforming and non-conforming numerical schemes for diffusion equations. We develop here this framework for the approximation of the steady state and transient incompressible Stokes equations with homogeneous Dirichlet boundary conditions. Using this framework, we establish generic convergence results-by error estimates in the case of the steady problem, and by compactness arguments in the case of the transient problem-that are applicable to both old and new schemes for Stokes' equations. Three classical methods (MAC, Taylor-Hood and Crouzeix-Raviart schemes) are shown to fit into the gradient schemes framework; some of the convergence results obtained for those through the framework are new. We also show that a Hybrid Mixed Mimetic scheme, extension of the Crouzeix-Raviart scheme to any polyhedral mesh, can be designed within the gradient scheme framework; this scheme is new for Stokes' equations, and our abstract analysis establishes its convergence along with error estimates.

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Section: The Linear Mapping DIVmentioning

confidence: 99%

Gradient Schemes for Stokes problem

Droniou¹,

Eymard

Feron

2015

IMA J Numer Anal

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“…This property has been used in [17] to build a one-parameter family of symmetric positive definite matrices o for a triangle. As a particular case, the family includes the mass matrix appearing in the finite element discretizations with the Raviart-Thomas finite elements.…”

Section: S2 (Consistency Ofmentioning

confidence: 99%

Convergence of the Mimetic Finite Difference Method for Diffusion Problems on Polyhedral Meshes

Brezzi¹,

Lipnikov²,

Shashkov³

2005

SIAM J. Numer. Anal.

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325

379

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“…For triangular cells. In [16], the authors perform a detailed analysis of the SOM for triangular cells. They show that their scheme preserves the linear solutions provided the corner volumes are equal to the third of the cell area.…”

Section: Comparison With the Discretization Based On The Sommentioning

confidence: 99%