Covolume-upwind finite volume approximations for linear elliptic partial differential equations

Ju, Lili; Li, Tian; Xiao, Xiao; Zhang, Weidong

doi:10.1016/j.jcp.2012.05.004

Cited by 4 publications

(5 citation statements)

References 29 publications

(34 reference statements)

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“…We would also like to note that the condition numbers of the systems produced by our proposed FEM‐WUFVM are roughly

O (h^{- 1})

for convection‐dominated cases

D = 10^{- 3}

and

D = 10^{- 8}

, which are the same order but much smaller than the ones obtained by FEM‐UFVM. However, the condition numbers produced by the covolume‐upwind finite volume method proposed in are more than

O (h^{- 2})

for these cases. See Fig.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Then we define the primary grid

T_{h}^{Q}

Ω

, i.e., the set of rectangular cells

{Ω_{i j} | Ω_{i j} = [x_{i - 1}, x_{i}] \times [y_{j - 1}, y_{j}], i = 1, \dots, N_{x}, j = 1, \dots, N_{y}} .

Similar to Refs. , we define our finite volume approximation scheme for the convection term shown in problem on a nonstandard dual grid. The dual partition for finite volume approximation is described at below.…”

Section: Model Problem and Domain Partitionsmentioning

confidence: 99%

“…Recently, a high‐order accuracy weighted upwind control volume method for convection‐diffusion problems was proposed in Refs. . Based on the physical feathers of the problems, a nonstandard control volume dual grid was first defined by locally shifting the standard dual grid in upstream direction.…”

Section: Introductionmentioning

confidence: 99%

“…Different from the method in Refs. , the proposed method falls into the standard finite element variational framework, and does not need the information of the neighboring elements, when the finite volume discretization is formed in each individual element of the dual grid. By carrying out the numerical analysis and various numerical experiments, we demonstrate that our newly proposed combined finite element‐weighted upwind FVM (FEM‐WUFVM) is not only numerically stable but also optimally convergent, even for strongly convection‐dominated problems.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A new combined finite element-upwind finite volume method for convection-dominated diffusion problems

Wang

Sun

2015

Numer. Methods Partial Differential Eq.

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In this article, we develop a combined finite element-weighted upwind finite volume method for convectiondominated diffusion problems in two dimensions, which discretizes the diffusion term with the standard finite element scheme, and the convection and source terms with the weighted upwind finite volume scheme. The developed method leads to a totally new scheme for convection-dominated problems, which overcomes numerical oscillation, avoids numerical dispersion, and has high-order accuracy. Stability analyses of the scheme are given for the problems with constant coefficients. Numerical experiments are presented to illustrate the stability and optimal convergence of our proposed method.

show abstract

“…We would also like to note that the condition numbers of the systems produced by our proposed FEM‐WUFVM are roughly

O (h^{- 1})

for convection‐dominated cases

D = 10^{- 3}

and

D = 10^{- 8}

, which are the same order but much smaller than the ones obtained by FEM‐UFVM. However, the condition numbers produced by the covolume‐upwind finite volume method proposed in are more than

O (h^{- 2})

for these cases. See Fig.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Then we define the primary grid

T_{h}^{Q}

Ω

, i.e., the set of rectangular cells

{Ω_{i j} | Ω_{i j} = [x_{i - 1}, x_{i}] \times [y_{j - 1}, y_{j}], i = 1, \dots, N_{x}, j = 1, \dots, N_{y}} .

Section: Model Problem and Domain Partitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A new combined finite element-upwind finite volume method for convection-dominated diffusion problems

Wang

Sun

2015

Numer. Methods Partial Differential Eq.

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show abstract

“…Thereby, in order to deal with anisotropic diffusion tensors, Hanz Martin Cheng et al establish a generalised complete flux scheme (GCFS) by combining CFS and the hybrid mimetic mixed method [39]. For FVEM, some schemes on rectangular meshes were proposed by constructing special dual meshes [40][41][42]. However, constructing special dual elements according to Peclet number needs additional computational cost.…”

Section: Introductionmentioning

confidence: 99%

A New Upwind Finite Volume Element Method for Convection-Diffusion-Reaction Problems on Quadrilateral Meshes

Li,

Yang,

null

et al. 2024

CiCP

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This paper is devoted to constructing and analyzing a new upwind finite volume element method for anisotropic convection-diffusion-reaction problems on general quadrilateral meshes. We prove the coercivity, and establish the optimal error estimates in H 1 and L 2 norm respectively. The novelty is the discretization of convection term, which takes the two terms Taylor expansion. This scheme has not only optimal first-order accuracy in H 1 norm, but also optimal second-order accuracy in L 2 norm, both for dominant diffusion and dominant convection. Numerical experiments confirm the theoretical results.

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The second-order modified upwind PPM characteristic difference method and analysis for solving convection-diffusion equations

Ren,

Zhang,

Zhou

2024

Numer Algor

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Covolume-upwind finite volume approximations for linear elliptic partial differential equations

Cited by 4 publications

References 29 publications

A new combined finite element-upwind finite volume method for convection-dominated diffusion problems

A new combined finite element-upwind finite volume method for convection-dominated diffusion problems

A New Upwind Finite Volume Element Method for Convection-Diffusion-Reaction Problems on Quadrilateral Meshes

The second-order modified upwind PPM characteristic difference method and analysis for solving convection-diffusion equations

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