Multiscale enrichment of a finite volume element method for the stationary Navier–Stokes problem

Wen, Juan; He, Ya‐Ling; Yang, Jianhong

doi:10.1080/00207160.2013.768765

Cited by 6 publications

(1 citation statement)

References 24 publications

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“…The FVM has many advantages that belong to finite difference and FEMs, such as, it is easy to set up and implement, conserve mass locally and the FVM also can treat the complicated geometry and general boundary conditions flexibly. For the incompressible flow, we can refer to the combination of FVM with stabilized method [2,36,39], with multiscale enrichment method [50,51], with adaptive mesh [33], with the immersed-boundary method [43] for the Navier-Stokes equations, [40,41] for the viscoelastic equations and [42] for the Boussinesq equations et al For the incompressible MHD equations, several efficient FVMs have been designed and implemented in recent years, such as the divergence-free semi-implicit FVM [18], the finite volume spectral element method [45], the high-order central essentially nonoscillatory (ENO) scheme [47], the divergence-free weighted essentially nonoscillatory (WENO) reconstruction-based scheme [55] and the references therein. Due to the nonlinearity and coupling of variables, up to the knowledge of Web of Science, the published papers of FVM for the incompressible MHD equations mainly concentrated on the design of numerical schemes, while the rigorous theoretical analysis results are still lacking.…”

Section: Introductionmentioning

confidence: 99%

Two‐level stabilized finite volume method for the stationary incompressible magnetohydrodynamic equations

Chu

Chen

Zhang

2023

Numerical Methods Partial

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In this paper, a two‐level stabilized finite volume method is developed and analyzed for the steady incompressible magnetohydrodynamic (MHD) equations. The linear polynomial space is used to approximate the velocity, pressure and magnetic fields, and two local Gauss integrations are introduced to overcome the restriction of discrete inf‐sup condition. Firstly, the existence and uniqueness of the solution of the discrete problem in the stabilized finite volume method are proved by using the Brouwer's fixed point theorem. ‐stability results of numerical solutions are also presented. Secondly, optimal error estimates of numerical solutions in and ‐norms are established by using the energy method and constructing the corresponding dual problem. Thirdly, the stability and convergence of two‐level stabilized finite volume method for the stationary incompressible MHD equations are provided. Theoretical findings show that the two‐level method has the same accuracy as the one‐level method with the mesh sizes . Finally, some numerical results are provided to identify with the established theoretical findings and show the performances of the considered numerical schemes.

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Section: Introductionmentioning

confidence: 99%

Two‐level stabilized finite volume method for the stationary incompressible magnetohydrodynamic equations

Chu

Chen

Zhang

2023

Numerical Methods Partial

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Two‐level multiscale enrichment finite volume method based on the Newton iteration for the stationary incompressible magnetohydrodynamics flow

Shen,

Chen,

Bian

et al. 2023

Math Methods in App Sciences

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In this paper, we propose a stabilized finite volume method based on the Newton's type iteration for the steady incompressible magnetohydrodynamics (MHD) problem. In order to reduce the computational complexity, the lowest order mixed finite element pair ( ‐ ‐ ) is adopted to approximate the velocity, pressure and magnetic fields, and the multiscale enrichment method is introduced to overcome the restriction of discrete inf‐sup (LBB) condition. We firstly solve the incompressible MHD equations by the Newton iterations on a coarse mesh with the mesh size , stability and convergence results of numerical solutions are provided. Secondly, the combination of the two‐level method with the Newton iteration is used to approximate the considered problem, a coupled nonlinear problem is solved on the coarse mesh , then the Stokes and Maxwell equations are considered on a fine mesh with the mesh size , the uniform stability and optimal error estimates of two‐level Newton iterative method are provided. Theoretical findings show that the two‐level method has the same order as the one‐level method in ‐norm as long as the mesh sizes satisfy . However, the two‐level method involves much less work than the one‐level method. Finally, some numerical examples are presented to demonstrate the effectiveness of the considered numerical schemes.

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A stabilized finite volume method for the stationary Navier–Stokes equations

Sheng

Zhang

Jiang

2016

Chaos, Solitons & Fractals

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Multiscale enrichment of a finite volume element method for the stationary Navier–Stokes problem

Cited by 6 publications

References 24 publications

Two‐level stabilized finite volume method for the stationary incompressible magnetohydrodynamic equations

Two‐level stabilized finite volume method for the stationary incompressible magnetohydrodynamic equations

Two‐level multiscale enrichment finite volume method based on the Newton iteration for the stationary incompressible magnetohydrodynamics flow

A stabilized finite volume method for the stationary Navier–Stokes equations

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