On the second-order convergence of a function reconstructed from finite volume approximations of the Laplace equation on Delaunay-Voronoi meshes

Omnes, Pascal

doi:10.1051/m2an/2010068

Cited by 9 publications

(4 citation statements)

References 33 publications

(48 reference statements)

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“…It should also be mentioned that some post-processing techniques can provide, under certain circumstances, an O(h 2 ) convergence in L 2 norm for functions reconstructed from the solutions to finite volume approximations. One of these postprocessing technique, using two TPFA schemes on two dual meshes, is described in [37]. These quadratic convergences of post-processed solutions however do not say anything specific on the super-convergence of the original finite volume scheme.…”

Section: Introductionmentioning

confidence: 99%

Improved $L^2$ estimate for gradient schemes and super-convergence of the TPFA finite volume scheme

Droniou

Nataraj

2017

IMA Journal of Numerical Analysis

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The gradient discretisation method is a generic framework that is applicable to a number of schemes for diffusion equations, and provides in particular generic error estimates in L 2 and H 1 -like norms. In this paper, we establish an improved L 2 error estimate for gradient schemes. This estimate is applied to a family of gradient schemes, namely, the Hybrid Mimetic Mixed (HMM) schemes, and yields an O(h 2 ) super-convergence rate in L 2 norm, provided local compensations occur between the cell points used to define the scheme and the centers of mass of the cells. To establish this result, a modified HMM method is designed by just changing the quadrature of the source term; this modified HMM enjoys a super-convergence result even on meshes without local compensations. Finally, the link between HMM and Two-Point Flux Approximation (TPFA) finite volume schemes is exploited to partially answer a long-standing conjecture on the super-convergence of TPFA schemes.Proof. The proof hinges on two tricks. In Step 1, letting u D * be the solution to the modified HMM scheme, we show thatCombined with the result from Step 1 and the super-convergence property (4.8) of the modified HMM method, this concludes the proof.Step 1: Comparison of the solutions to the schemes for D and D * . Let u D * be the solution to (2.1) with D * instead of D. Subtracting the two gradient schemes corresponding to D * and D we see that, for all v D ∈ X D,0 , Ω

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Section: Introductionmentioning

confidence: 99%

Improved $L^2$ estimate for gradient schemes and super-convergence of the TPFA finite volume scheme

Droniou

Nataraj

2017

IMA Journal of Numerical Analysis

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“…We obtain a second order convergence for the L 2 -norm of the velocity. This super-convergence of the L 2 -norm is classical for finite volume method, however its proof still remains an open problem see [32].…”

Section: S Krellmentioning

confidence: 98%

Stabilized DDFV schemes for stokes problem with variable viscosity on general 2D meshes

Krell¹

2010

Numer. Methods Partial Differential Eq.

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Abstract. "Discrete Duality Finite Volume" schemes (DDFV for short) on general meshes are studied here for Stokes problems with variable viscosity with Dirichlet boundary conditions. The aim of this work is to analyze the well-posedness of the scheme and its convergence properties. The DDFV method requires a staggered scheme, the discrete unknowns, the components of the velocity and the pressure, are located on different nodes. The scheme is stabilized using a finite volume analogue to Brezzi-Pitkäranta techniques. This scheme is proved to be well-posed on general meshes and to be first order convergent in a discrete H 1 -norm and a discrete L 2 -norm for respectively the velocity and the pressure. Finally numerical experiments confirm the theoretical prediction, in particular on locally refined non conformal meshes.

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“…The DDFV methods come in two formulations. The first formulation is based on interface flux computations for primary and dual meshes, accounting with the interface flux continuity (see, e.g., [20,21]) and the second formulation of DDFV is based on pressure gradient reconstructions over a diamond grid (see [22][23][24]). Note that this second formulation attracted the attention of some mathematicians as Andreianov, Boyer, and Hubert who have greatly contributed to its mathematical development.…”

Section: Introduction and The Model Problemmentioning

confidence: 99%

Convergence Analysis on Unstructured Meshes of a DDFV Method for Flow Problems with Full Neumann Boundary Conditions

Jeutsa

Njifenjou

Nganhou

2016

Journal of Applied Mathematics

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A Discrete Duality Finite Volume (DDFV) method to solve on unstructured meshes the flow problems in anisotropic nonhomogeneous porous media with full Neumann boundary conditions is proposed in the present work. We start with the derivation of the discrete problem. A result of existence and uniqueness of a solution for that problem is given thanks to the properties of its associated matrix combined with adequate assumptions on data. Their theoretical properties, namely, stability and error estimates (in discrete energy norms andL2-norm), are investigated. Numerical test is provided.

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On the second-order convergence of a function reconstructed from finite volume approximations of the Laplace equation on Delaunay-Voronoi meshes

Cited by 9 publications

References 33 publications

Improved $L^2$ estimate for gradient schemes and super-convergence of the TPFA finite volume scheme

Improved $L^2$ estimate for gradient schemes and super-convergence of the TPFA finite volume scheme

Stabilized DDFV schemes for stokes problem with variable viscosity on general 2D meshes

Convergence Analysis on Unstructured Meshes of a DDFV Method for Flow Problems with Full Neumann Boundary Conditions

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