Convergence analysis of a finite volume method via a new nonconforming finite element method

Vanselow, Reiner; Scheffler, Hans–Peter

doi:10.1002/(sici)1098-2426(199803)14:2<213::aid-num5>3.0.co;2-r

Cited by 13 publications

(24 citation statements)

References 11 publications

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“…Now the set (24), (25), (26) and (27) provides for I + K + J Γ + 2 equations, while there are only I + K + J Γ unknowns. But this is no matter, since we may prove the following proposition Proposition 3.5.…”

Section: The Case Of Neumann Boundary Conditionsmentioning

confidence: 99%

“…But this is no matter, since we may prove the following proposition Proposition 3.5. Equations (24), (25) and (26) provide only for I + K + J Γ − 2 independent equations and the set (24), (25), (26) and (27) has a unique solution.…”

Section: The Case Of Neumann Boundary Conditionsmentioning

confidence: 99%

“…If the line joining the centers of two adjacent cells is perpendicular to the corresponding interface, then the value of ∇φ · n on this interface may simply be computed by a finite difference. This is the case for the so-called "admissible meshes" described in [9] which include Voronoi-type meshes (see also [20,25]). But for other general meshes (in particular for non-conforming meshes which can never be "admissible"), a possibility is to reconstruct the whole gradient, and not only its normal component (one may find alternative approaches to the reconstruction of the whole gradient in [11,19]).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids

Domelevo¹,

Omnes²

2005

ESAIM: M2AN

177

251

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Abstract. We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires "Voronoi-type" meshes. We show the equivalence of this finite volume method with a non-conforming finite element method with basis functions being P 1 on the cells, generally called "diamond-cells", of a third mesh. Under geometrical conditions on these diamondcells, we prove a first-order convergence both in the H 1 0 norm and in the L 2 norm. Superconvergence results are obtained on certain types of homothetically refined grids. Finally, numerical experiments confirm these results and also show second-order convergence in the L 2 norm on general grids. They also indicate that this method performs particularly well for the approximation of the gradient of the solution, and may be used on degenerating triangular grids. An example of application on nonconforming locally refined grids is given.Mathematics Subject Classification. 35J05, 35J25, 65N12, 65N15, 65N30.

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Section: The Case Of Neumann Boundary Conditionsmentioning

confidence: 99%

Section: The Case Of Neumann Boundary Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids

Domelevo¹,

Omnes²

2005

ESAIM: M2AN

177

251

View full text Add to dashboard Cite

show abstract

“…9.1), in the discrete L 2 norm as well, [22], Theorem 9.4. Related results in the energy norm have been shown in [32,36]. On the other hand, deriving necessary and/or sufficient conditions to obtain second-order accuracy in the discrete L 2 norm is still an open issue on general admissible meshes.…”

Section: P Omnesmentioning

confidence: 99%

“…Moreover, the error estimation (3.8) is inferred from [19], Theorem 5.20, in which the angle τ * is always equal to π/2 in the special case of Delaunay-Voronoi meshes. Related convergence results in a discrete norm for each component (primal or dual) of the gradient may be found in [22,32,36].…”

Section: Equation (310) May Be Rewritten Asmentioning

confidence: 99%

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

Omnes

2010

ESAIM: M2AN

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Abstract. Cell-centered and vertex-centered finite volume schemes for the Laplace equation with homogeneous Dirichlet boundary conditions are considered on a triangular mesh and on the Voronoi diagram associated to its vertices. A broken P 1 function is constructed from the solutions of both schemes. When the domain is two-dimensional polygonal convex, it is shown that this reconstruction converges with second-order accuracy towards the exact solution in the L 2 norm, under the sufficient condition that the right-hand side of the Laplace equation belongs to H 1 (Ω).Mathematics Subject Classification. 65N15, 65N30, 35J05.

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A mixed finite volume method for elliptic problems

Mishev

Chen

2007

Numerical Methods Partial

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We derive a novel finite volume method for the elliptic equation, using the framework of mixed finite element methods to discretize the pressure and velocities on two different grids (covolumes), triangular (tetrahedral) mesh and control volume mesh. The new discretization is defined for tensor diffusion coefficient and well suited for heterogeneous media. When the control volumes are created by connecting the center of gravity of each triangle to the midpoints of its edges, we show that the discretization is stable and first order accurate for both scalar and vector unknowns.

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Convergence analysis of a finite volume method via a new nonconforming finite element method

Cited by 13 publications

References 11 publications

A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids

A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids

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

A mixed finite volume method for elliptic problems

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