A stabilized finite element method based on two local Gauss integrations for a coupled Stokes–Darcy problem

Li, Rui; Li, Jian; Chen, Zhangxin

doi:10.1016/j.cam.2015.06.014

Cited by 46 publications

(31 citation statements)

References 32 publications

(47 reference statements)

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“…The parameter β is a positive stabilization parameter, and its value is chosen by trials. In , β = 1.0 is selected in their numerical experiments. In our numerical examples, β = 1.0, 0, 0.1, 0.01, and 0.001 are tested.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Lemma (Theorem ). Assume that the constant α in BJS interface condition is sufficiently small, then the generalized bilinear form

\overset{true~}{B} (((),,, {boldu}_{h}, ϕ_{h}, p_{h}), ((),,, {boldv}_{h}, ψ_{h}, q_{h}))

satisfies the continuity

\overset{true~}{B} (((),,, {boldu}_{h}, ϕ_{h}, p_{h}), ((),,, {boldv}_{h}, ψ_{h}, q_{h})) \leq C ({(‖‖, {boldu}_{h})}_{H_{f}} + {(‖‖, ϕ_{h})}_{H_{p}} + {(‖‖, p_{h})}_{L^{2} (Ω_{f})}^{2}) ({(‖‖, {boldv}_{h})}_{H_{f}} + {(‖‖, ψ_{h})}_{H_{p}} + {(‖‖, q_{h})}_{L^{2} (Ω_{f})}^{2})

and the coercive property

\sup_{(v_{h}, ϕ_{h}, q_{h}) \in (H_{fh}, H_{ph}, Q_{h})} \frac{∣ \overset{true~}{B} (((),,, {boldu}_{h}, ϕ_{h}, p_{h}), ((),,, {boldv}_{h}, ψ_{h}, q_{h})) ∣}{{({(‖‖, {boldv}_{h})}_{H_{f}}^{2} + {(‖‖, ψ_{h})}_{H_{p}}^{2} + {(‖‖, q_{h})}_{L^{2} (Ω_{f})}^{2})}^{\frac{1}{2}}} \geq γ ((), ‖u_{}‖)

…”

Section: The Error Analysismentioning

confidence: 99%

“…These phenomena are modeled by various partial differential equations such as the Navier–Stokes/Darcy equations, and the Stokes/Darcy equations. Many coupling methods for these models are studied in recent years .…”

Section: Introductionmentioning

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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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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Section: Numerical Resultsmentioning

confidence: 99%

“…Lemma (Theorem ). Assume that the constant α in BJS interface condition is sufficiently small, then the generalized bilinear form

\overset{true~}{B} (((),,, {boldu}_{h}, ϕ_{h}, p_{h}), ((),,, {boldv}_{h}, ψ_{h}, q_{h}))

satisfies the continuity

\overset{true~}{B} (((),,, {boldu}_{h}, ϕ_{h}, p_{h}), ((),,, {boldv}_{h}, ψ_{h}, q_{h})) \leq C ({(‖‖, {boldu}_{h})}_{H_{f}} + {(‖‖, ϕ_{h})}_{H_{p}} + {(‖‖, p_{h})}_{L^{2} (Ω_{f})}^{2}) ({(‖‖, {boldv}_{h})}_{H_{f}} + {(‖‖, ψ_{h})}_{H_{p}} + {(‖‖, q_{h})}_{L^{2} (Ω_{f})}^{2})

and the coercive property

\sup_{(v_{h}, ϕ_{h}, q_{h}) \in (H_{fh}, H_{ph}, Q_{h})} \frac{∣ \overset{true~}{B} (((),,, {boldu}_{h}, ϕ_{h}, p_{h}), ((),,, {boldv}_{h}, ψ_{h}, q_{h})) ∣}{{({(‖‖, {boldv}_{h})}_{H_{f}}^{2} + {(‖‖, ψ_{h})}_{H_{p}}^{2} + {(‖‖, q_{h})}_{L^{2} (Ω_{f})}^{2})}^{\frac{1}{2}}} \geq γ ((), ‖u_{}‖)

…”

Section: The Error Analysismentioning

confidence: 99%

See 1 more Smart Citation

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

Jia

2018

Numerical Methods Partial

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show abstract

“…Unlike in [10] where a combination of a generalized mini element and local projection (LPS) is analyzed and in [14] where a method based on two local Gauss integrals for the Stokes equations is used, here we will analyze the problem using a subgrid method [12,15,16]. For this method the filter, with respect to the global Lagrange interpolant 2ℎ , onto a coarser mesh T 2ℎ is used.…”

Section: Finite Element Discretizationmentioning

confidence: 99%

Analysis of Subgrid Stabilization Method for Stokes-Darcy Problems

Nafa

2016

Advances in Numerical Analysis

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A number of techniques, used as remedy to the instability of the Galerkin finite element formulation for Stokes like problems, are found in the literature. In this work we consider a coupled Stokes-Darcy problem, where in one part of the domain the fluid motion is described by Stokes equations and for the other part the fluid is in a porous medium and described by Darcy law and the conservation of mass. Such systems can be discretized by heterogeneous mixed finite elements in the two parts. A better method, from a computational point of view, consists in using a unified approach on both subdomains. Here, the coupled Stokes-Darcy problem is analyzed using equal-order velocity and pressure approximation combined with subgrid stabilization. We prove that the obtained finite element solution is stable and converges to the classical solution with optimal rates for both velocity and pressure.

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“…(3) Compared with the standard finite element method, its analyses of H 1 -norm and L 2 -norm for the velocity and pressure are derived without any high order regularity assumptions on the exact solution. So this method has been widely used to consider various kinds of problems [17,18,25,28,36]. In this paper, based on the regular triangular partitions of domain and the pressure project method, we consider the stabilized adaptive finite volume method for the Oseen equations by the lowest equal order element (i.e., P 1 -P 1 pair).…”

Section: Introductionmentioning

confidence: 99%

Adaptive stabilized finite volume method and convergence analysis for the Oseen equations

Zhang

2018

Bound Value Probl

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In this paper, based on the pressure project method, we consider an adaptive stabilized finite volume method for the Oseen equations with the lowest equal order finite element pair. Firstly, we develop the discrete forms in both finite element and finite volume methods, and establish the existence and uniqueness of numerical solutions by establishing the equivalence of linear terms in finite element and finite volume methods. Secondly, a residual type a posteriori error estimator is designed, and the computable global upper and local lower bounds between the exact solutions and the finite volume solutions are established. Thirdly, a discrete local lower bound between two successive finite volume solutions is obtained, convergence analysis of the adaptive stabilized finite volume method is also performed. Finally, some numerical results are presented to verify the performances of the developed error estimators and confirm the established theoretical findings. MSC: 65N15; 65N30; 76D05

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A stabilized finite element method based on two local Gauss integrations for a coupled Stokes–Darcy problem

Cited by 46 publications

References 32 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

Analysis of Subgrid Stabilization Method for Stokes-Darcy Problems

Adaptive stabilized finite volume method and convergence analysis for the Oseen equations

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