A Decoupling Method with Different Subdomain Time Steps for the Non-Stationary Navier-Stokes ⁄ Darcy Model

Pei-lin, Shi; Li, Kaitai; Jia, H. M.; Jia, Hongwei

doi:10.4208/jcm.1606-m2015-0436

Cited by 8 publications

(6 citation statements)

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“…Here h denotes the mesh size. To solve our problem, we use the following finite element subspace W h = H fh × H ph ⊂ W and Q h ⊂ Q :

\begin{array}{l} H_{fh} = ({};,, v_{h} \in H_{f} \cap C^{0} {(\overset{Ω}{true})}^{d}, v_{h} ∣_{K} \in P_{1} {(K)}^{d}, \forall K \in τ_{h}), \\ H_{ph} = ({};,, ψ_{h} \in H_{p} \cap C^{0} ((), \overset{true‾}{normalΩ}), ψ_{h} ∣_{K} \in P_{1} ((), K), \forall K \in τ_{h}), \\ Q_{h} = ({};,, q_{h} \in Q \cap C^{0} ((), \overset{true‾}{normalΩ}), q_{h} ∣_{K} \in P_{1} ((), K), \forall K \in τ_{h}) . \end{array}

…”

Section: The Decoupled Stabilized Characteristic Finite Element Methodsmentioning

confidence: 99%

“…Here h denotes the mesh size. To solve our problem, we use the following finite element subspace W h = H fh × H ph ⊂ W and Q h ⊂ Q [11]:…”

Section: The Decoupled Stabilized Characteristic Finite Element Methodsmentioning

confidence: 99%

“…In this section, we will derive the error estimate of the CSFEM. First, we assume that the solution ( u , p , ϕ ) of problems – satisfies the following regularity hypotheses :

\begin{array}{l} ((), 1) boldu \in L^{\infty} ((),,, 0, T, H^{2} {(Ω_{f})}^{2}), ϕ \in L^{\infty} ((),,, 0, T, H^{2} ((), {normalΩ}_{p})), \\ ((), 2) u_{t} \in L^{2} ((),,, 0, T, H^{2} {(Ω_{f})}^{2}), ϕ_{t} \in L^{2} ((),,, 0, T, H^{2} ((), {normalΩ}_{p})), \\ ((), 3) p \in L^{\infty} ((),,, 0, T, H^{1} ((), {normalΩ}_{f})), p_{t} \in L^{2} ((),,, 0, T, H^{1} ((), {normalΩ}_{f})) . \end{array}

…”

Section: The Error Analysismentioning

confidence: 99%

“…In this section, we will derive the error estimate of the CSFEM. First, we assume that the solution (u, p, ) of problems (2.1)-(2.6) satisfies the following regularity hypotheses [11]:…”

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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“…Here h denotes the mesh size. To solve our problem, we use the following finite element subspace W h = H fh × H ph ⊂ W and Q h ⊂ Q :

\begin{array}{l} H_{fh} = ({};,, v_{h} \in H_{f} \cap C^{0} {(\overset{Ω}{true})}^{d}, v_{h} ∣_{K} \in P_{1} {(K)}^{d}, \forall K \in τ_{h}), \\ H_{ph} = ({};,, ψ_{h} \in H_{p} \cap C^{0} ((), \overset{true‾}{normalΩ}), ψ_{h} ∣_{K} \in P_{1} ((), K), \forall K \in τ_{h}), \\ Q_{h} = ({};,, q_{h} \in Q \cap C^{0} ((), \overset{true‾}{normalΩ}), q_{h} ∣_{K} \in P_{1} ((), K), \forall K \in τ_{h}) . \end{array}

…”

Section: The Decoupled Stabilized Characteristic Finite Element Methodsmentioning

confidence: 99%

“…Here h denotes the mesh size. To solve our problem, we use the following finite element subspace W h = H fh × H ph ⊂ W and Q h ⊂ Q [11]:…”

Section: The Decoupled Stabilized Characteristic Finite Element Methodsmentioning

confidence: 99%

“…In this section, we will derive the error estimate of the CSFEM. First, we assume that the solution ( u , p , ϕ ) of problems – satisfies the following regularity hypotheses :

\begin{array}{l} ((), 1) boldu \in L^{\infty} ((),,, 0, T, H^{2} {(Ω_{f})}^{2}), ϕ \in L^{\infty} ((),,, 0, T, H^{2} ((), {normalΩ}_{p})), \\ ((), 2) u_{t} \in L^{2} ((),,, 0, T, H^{2} {(Ω_{f})}^{2}), ϕ_{t} \in L^{2} ((),,, 0, T, H^{2} ((), {normalΩ}_{p})), \\ ((), 3) p \in L^{\infty} ((),,, 0, T, H^{1} ((), {normalΩ}_{f})), p_{t} \in L^{2} ((),,, 0, T, H^{1} ((), {normalΩ}_{f})) . \end{array}

…”

Section: The Error Analysismentioning

confidence: 99%

“…In this section, we will derive the error estimate of the CSFEM. First, we assume that the solution (u, p, ) of problems (2.1)-(2.6) satisfies the following regularity hypotheses [11]:…”

Section: The Error Analysismentioning

confidence: 99%

See 2 more Smart Citations

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

Jia

2018

Numerical Methods Partial

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show abstract

“…Compare to Navier-Stokes model, the convection term causes some difficulties in theoretical analysis and numerical simulation. To deal with convection term, the assumption u • n f > 0 is used [6,9]. Our proposed method not only relax the influence of parameters but also improves the mass conservation.…”

mentioning

confidence: 99%

Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model

Wang

Meng

Jia

2020

era

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In this paper, we construct a modular grad-div stabilization method for the Navier-Stokes/Darcy model, which is based on the first order Backward Euler scheme. This method does not enlarge the accuracy of numerical solution, but also can improve mass conservation and relax the influence of parameters. Herein, we give stability analysis and error estimations. Finally, by some numerical experiment, the scheme our proposed is shown to be valid.

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Diffuse interface modeling of non‐isothermal Stokes‐Darcy flow with immersed transmissibility conditions

Suh

2024

Numerical Meth Engineering

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The coupling between free and porous medium flows has received significant attention since it plays an important role in a wide range of problems from fluid‐soil interactions to biofluid dynamics. However, modeling this coupled process remains a difficult task as it often involves a domain decomposition algorithm in conjunction with a special treatment at the interface. The problem can become more challenging under non‐isothermal conditions because it requires the iterative procedure at every time step to simultaneously meet the transient mass continuity, force equilibrium, and energy balance for the entire system. This article presents a diffuse interface framework for modeling non‐isothermal Stokes‐Darcy flow and the corresponding finite element formulation that bypasses the need for explicitly splitting the domain into two, which enables the unified treatment for distinct regions with different hydrothermal flow regimes. To achieve this goal, we employ the Allen‐Cahn type phase field model to generate the diffuse geometry, where the solution field can be seen as a regularized approximation of the Heaviside indicator function, allowing us to transfer the interface conditions into a set of immersed boundary conditions. Our formulation suggests that the isothermal operator splitting strategy can be adopted without compromising accuracy if the heat and mass transfer processes are decoupled by assuming that the density and viscosity of the phase constituents are independent to the temperature. Numerical examples are also introduced to verify the implementation and to demonstrate the model capacity.

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A Decoupling Method with Different Subdomain Time Steps for the Non-Stationary Navier-Stokes ⁄ Darcy Model

Cited by 8 publications

References 0 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

Numerical analysis of modular grad-div stability methods for the time-dependent Navier-Stokes/Darcy model

Diffuse interface modeling of non‐isothermal Stokes‐Darcy flow with immersed transmissibility conditions

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