Optimal estimates on stabilized finite volume methods for the incompressible Navier–Stokes model in three dimensions

Li, Jian; Lin, Xiaolin; Zhao, Xin

doi:10.1002/num.22294

Cited by 7 publications

(4 citation statements)

References 41 publications

(112 reference statements)

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“…Then, the weak formulation of the dual-porosity-Navier-Stokes model (2.1)-(2.12) as follows [31][32][33][34][35][36]. we suppose ⃗ f s ∈ H −1 (Ω s ) and f d ∈ L 2 (Ω d ).…”

Section: Preliminaries Weak Formulations and Finite Element Spacesmentioning

confidence: 99%

“…Trace inequality: There exists a positive constant C T depending on the domain Ω s such that for all

{\overset{v}{true}}_{s} \in Y_{s}

, we have

‖ {\overset{v}{true}}_{s} ‖_{L^{2} false(double-struckI false)} \leq C_{T} ‖ {\overset{v}{true}}_{s} ‖_{0}^{\frac{1}{2}} ‖ \nabla v_{s} ‖_{0}^{\frac{1}{2}} .

Then, the weak formulation of the dual‐porosity‐Navier–Stokes model (2.1)–(2.12) as follows [31–36]. we suppose

{\overset{f}{true}}_{s} \in H^{- 1} false(Ω_{s} false)

and f d ∈ L 2 (Ω d ).…”

Section: Preliminaries Weak Formulations and Finite Element Spacesmentioning

confidence: 99%

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A decoupled stabilized finite element method for the dual‐porosity‐Navier–Stokes fluid flow model arising in shale oil

Gao

2020

Numerical Methods Partial

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In this paper, we consider the decoupled stabilized finite element method for the dual-porosity-Navier-Stokes model coupling the free flow region and the microfracture-matrix system by using four interface conditions on the interface. The stabilized finite element method is decoupled in two levels, and it allows the coupling problem to be divide into three subproblems in a non-iterative manner, which improves the computational efficiency. In addition, the stability and convergence of the decoupling scheme are also analyzed. Finally, the theoretical results are illustrated by some numerical experiments. KEYWORDS convergence, decoupled method, dual-porosity-Navier-Stokes, finite element method, numerical experiments, stability 1 INTRODUCTION Many scientists and engineers have investigated the fluid flow interaction between conduit and porous media region [1-3]. A large number of practical problems, such as groundwater flow system, interaction between surface, and groundwater flow, biochemical transportation, blood flow in artery, vein, and so forth [4-7], have been established relevant hydrodynamic models by scientists. In addition, a lot of efforts have been devoted to develop appropriate numerical methods to solve the Stokes-Darcy fluid system, including the coupled finite element methods [8-10], domain decomposition methods [11-13], the Lagrange multiplier method [14, 15], and so forth. Although the traditional Stokes-Darcy model [16-18] has been well studied, it has some limitations in describing the heterogeneity of porous media. It is worth noting that the real natural fractured

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“…Then, the weak formulation of the dual-porosity-Navier-Stokes model (2.1)-(2.12) as follows [31][32][33][34][35][36]. we suppose ⃗ f s ∈ H −1 (Ω s ) and f d ∈ L 2 (Ω d ).…”

Section: Preliminaries Weak Formulations and Finite Element Spacesmentioning

confidence: 99%

“…Trace inequality: There exists a positive constant C T depending on the domain Ω s such that for all

{\overset{v}{true}}_{s} \in Y_{s}

, we have

‖ {\overset{v}{true}}_{s} ‖_{L^{2} false(double-struckI false)} \leq C_{T} ‖ {\overset{v}{true}}_{s} ‖_{0}^{\frac{1}{2}} ‖ \nabla v_{s} ‖_{0}^{\frac{1}{2}} .

Then, the weak formulation of the dual‐porosity‐Navier–Stokes model (2.1)–(2.12) as follows [31–36]. we suppose

{\overset{f}{true}}_{s} \in H^{- 1} false(Ω_{s} false)

and f d ∈ L 2 (Ω d ).…”

Section: Preliminaries Weak Formulations and Finite Element Spacesmentioning

confidence: 99%

A decoupled stabilized finite element method for the dual‐porosity‐Navier–Stokes fluid flow model arising in shale oil

Gao

2020

Numerical Methods Partial

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“…The basic idea of this method was first proposed by Boland and Nicolaides, and has been vigorously developed since then. The recent work of Wen [ 10 ] and Li [ 11 ] paves the way for the numerical analysis of Stokes and Navier Stokes problems. In addition, He [ 12 ] and Li [ 13 ] have given the locally stable finite volume method for partial spatial discretization of Navier-Stokes problems, which has a good effect except some fully discrete results.…”

Section: Introductionmentioning

confidence: 99%

“…For this purpose, let’s assume

is the uniform and regular triangulation of

. It should be reminded that the finite element space here does not have the inf-sup condition for

, so a similar skill in the paper [ 11 , 12 , 13 ] is required. Because these papers mainly discuss the spatial discrete case, this paper studies a approximation based on time-discretization is Euler semi-implicit and space-discretization is

Locally stable FVM.…”

Section: Introductionmentioning

confidence: 99%

The Optimal Error Estimate of the Fully Discrete Locally Stabilized Finite Volume Method for the Non-Stationary Navier-Stokes Problem

Zhang

2022

Entropy

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This paper proves the optimal estimations of a low-order spatial-temporal fully discrete method for the non-stationary Navier-Stokes Problem. In this paper, the semi-implicit scheme based on Euler method is adopted for time discretization, while the special finite volume scheme is adopted for space discretization. Specifically, the spatial discretization adopts the traditional triangle P1−P0 trial function pair, combined with macro element form to ensure local stability. The theoretical analysis results show that under certain conditions, the full discretization proposed here has the characteristics of local stability, and we can indeed obtain the optimal theoretic and numerical order error estimation of velocity and pressure. This helps to enrich the corresponding theoretical results.

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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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Optimal estimates on stabilized finite volume methods for the incompressible Navier–Stokes model in three dimensions

Cited by 7 publications

References 41 publications

A decoupled stabilized finite element method for the dual‐porosity‐Navier–Stokes fluid flow model arising in shale oil

A decoupled stabilized finite element method for the dual‐porosity‐Navier–Stokes fluid flow model arising in shale oil

The Optimal Error Estimate of the Fully Discrete Locally Stabilized Finite Volume Method for the Non-Stationary Navier-Stokes Problem

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

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