Decoupled modified characteristics finite element method for the time dependent Navier–Stokes/Darcy problem

Si, Zhiyong; Li, Shishun

doi:10.1002/mma.2901

Cited by 19 publications

(10 citation statements)

References 34 publications

(50 reference statements)

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“…Assume a uniform time distribution with t m = m Δ t , m = 0, 1, 2, ···, M , where

Δ t = \frac{T}{M}

. We denote

{boldu}_{h}^{m}, p_{h}^{m}, ϕ_{h}^{m}

the discrete solution of the following scheme.Algorithm (CFEM) .…”

Section: The Decoupled Stabilized Characteristic Finite Element Methodsmentioning

confidence: 99%

“…To prove the error estimate, let us introduce some notations:

\begin{array}{l} u^{m} = boldu ((), t_{m}), ϕ^{m} = ϕ ((), t_{m}), p^{m} = p ((), t_{m}), \\ ((),, {\overset{u}{true}}^{m}, {\overset{true~}{ϕ}}^{m}) = ((),, P_{h}^{w} u^{m}, P_{h}^{w} ϕ^{m}), {\overset{true~}{p}}^{m} = P_{h}^{p} p^{m}, \\ e_{1}^{m} = {\overset{true~}{boldu}}^{m} - u^{m}, ϵ_{1}^{m} = {\overset{true~}{ϕ}}^{m} - ϕ^{m}, η_{1}^{m} = {\overset{true~}{p}}^{m} - p^{m}, \\ e^{m} = u_{h}^{m} - {\overset{true~}{boldu}}^{m}, ϵ^{m} = ϕ_{h}^{m} - {\overset{true~}{ϕ}}^{m}, η^{m} = p_{h}^{m} - {\overset{true~}{p}}^{m}, \end{array}

and recall a lemma to be used in our proof.Lemma (Lemma , ). Let

e (x, n) = [\frac{{boldu}^{m} (x) - {\overset{u}{true}}^{m - 1} (x)}{Δ t} -

(\frac{\partial u}{\partial t} (x, t_{m}) + ({boldu}^{m} ()))]

…”

Section: The Error Analysismentioning

confidence: 99%

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Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

Jia

2018

Numerical Methods Partial

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In this article, we propose and analyze a new decoupled characteristic stabilized finite element method for the time‐dependent Navier–Stokes/Darcy model. The key idea lies in combining the characteristic method with the stabilized finite element method to solve the decoupled model by using the lowest‐order conforming finite element space. In this method, the original model is divided into two parts: one is the nonstationary Navier–Stokes equation, and the other one is the Darcy equation. To deal with the difficulty caused by the trilinear term with nonzero boundary condition, we use the characteristic method. Furthermore, as the lowest‐order finite element pair do not satisfy LBB (Ladyzhen‐Skaya‐Brezzi‐Babuska) condition, we adopt the stabilized technique to overcome this flaw. The stability of the numerical method is first proved, and the optimal error estimates are established. Finally, extensive numerical results are provided to justify the theoretical analysis.

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“…Assume a uniform time distribution with t m = m Δ t , m = 0, 1, 2, ···, M , where

Δ t = \frac{T}{M}

. We denote

{boldu}_{h}^{m}, p_{h}^{m}, ϕ_{h}^{m}

the discrete solution of the following scheme.Algorithm (CFEM) .…”

Section: The Decoupled Stabilized Characteristic Finite Element Methodsmentioning

confidence: 99%

“…To prove the error estimate, let us introduce some notations:

\begin{array}{l} u^{m} = boldu ((), t_{m}), ϕ^{m} = ϕ ((), t_{m}), p^{m} = p ((), t_{m}), \\ ((),, {\overset{u}{true}}^{m}, {\overset{true~}{ϕ}}^{m}) = ((),, P_{h}^{w} u^{m}, P_{h}^{w} ϕ^{m}), {\overset{true~}{p}}^{m} = P_{h}^{p} p^{m}, \\ e_{1}^{m} = {\overset{true~}{boldu}}^{m} - u^{m}, ϵ_{1}^{m} = {\overset{true~}{ϕ}}^{m} - ϕ^{m}, η_{1}^{m} = {\overset{true~}{p}}^{m} - p^{m}, \\ e^{m} = u_{h}^{m} - {\overset{true~}{boldu}}^{m}, ϵ^{m} = ϕ_{h}^{m} - {\overset{true~}{ϕ}}^{m}, η^{m} = p_{h}^{m} - {\overset{true~}{p}}^{m}, \end{array}

and recall a lemma to be used in our proof.Lemma (Lemma , ). Let

e (x, n) = [\frac{{boldu}^{m} (x) - {\overset{u}{true}}^{m - 1} (x)}{Δ t} -

(\frac{\partial u}{\partial t} (x, t_{m}) + ({boldu}^{m} ()))]

…”

Section: The Error Analysismentioning

confidence: 99%

Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

Jia

2018

Numerical Methods Partial

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“…In [40], we presented modified characteristics gauge-Uzawa finite element method for the conduction-convection equation. In [42], the modified characteristics finite element method for the Navier-Stokes/Darcy problem was shown.…”

Section: Introductionmentioning

confidence: 99%

Modified Method of Characteristics Variational Multiscale Finite Element Method for Time Dependent Navier-Stokes Problems

Feng

2015

Mathematical Modelling and Analysis

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In this paper, a modified method of characteristics variational multiscale (MMOCVMS) finite element method is presented for the time dependent Navier-Stokes problems, which is leaded by combining the characteristics time discretization with the variational multiscale (VMS) finite element method in space. The theoretical analysis shows that this method has a good convergence property. In order to show the efficiency of the MMOCVMS finite element method, some numerical results of analytical solution problems are presented. First, we give some numerical results of lid-driven cavity flow with Re = 5000 and 7500 as the time is sufficient long. From the numerical results, we can see that the steady state numerical solutions of the time-dependent Navier-Stokes equations are obtained. Then, we choose Re = 10000, and we find that the steady state numerical solution is not stable from t = 200 to 300. Moreover, we also investigate numerically the flow around a cylinder problems. The numerical results show that our method is highly efficient.

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“…In [43], two decoupling algorithms for the steady Stokes-Darcy model based on two-grid discretizations were shown by Zhang and Yuan. In [39], we gave the decoupled modified characteristics finite element method for the time dependent Navier-Stokes/Darcy problem. The fullymixed finite element method for Stokes-Darcy problems was given by Gatica et al [8,24].…”

Section: Introductionmentioning

confidence: 99%

A Viscosity-Splitting Method for the Navier-Stokes/ Darcy Problem

Wang

2020

AAMM

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In this report, we give a viscosity splitting method for the Navier-Stokes/Darcy problem. In this method, the Navier-Stokes/Darcy equation is solved in three steps. In the first step, an explicit/ implicit formulation is used to solve the nonlinear problem. We introduce an artificial diffusion term θ∆u in our scheme whose purpose is to enlarge the time stepping and enhance numerical stability, especially for small viscosity parameter ν, by choosing suitable parameter θ. In the second step, we solve the Stokes equation for velocity and pressure. In the third step, we solve the Darcy equation for the piezometric head in the porous media domain. We use the numerical solutions at last time level to give the interface condition to decouple the Navier-Stokes equation and the Darcy's equation. The stability analysis, under some condition ∆t ≤ k 0 , k 0 > 0, is given. The error estimates prove our method has an optimal convergence rates. Finally, some numerical results are presented to show the performance of our algorithm.

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Decoupled modified characteristics finite element method for the time dependent Navier–Stokes/Darcy problem

Cited by 19 publications

References 34 publications

Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

Modified Method of Characteristics Variational Multiscale Finite Element Method for Time Dependent Navier-Stokes Problems

A Viscosity-Splitting Method for the Navier-Stokes/ Darcy Problem

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