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A new unified stabilized mixed finite element method of the Stokes-Darcy coupled problem: Isotropic discretization

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Pith's one-line read A single nonconforming Crouzeix-Raviart element, with normal-only continuity in the Darcy region, yields an optimally convergent stabilized mixed method for coupled Stokes-Darcy flow.

desk verdict Nice idea—cheaper unified Stokes–Darcy element—but the interpolation lemma for the enlarged space is unproved and the Fortin argument doesn't go through, so the convergence theorem is unsupported. read the letter →

arxiv 1908.01892 v1 pith:7UBUUBXD submitted 2019-08-05 math.NA cs.NA

Koffi Wilfrid Houédanou This is my paper · ORCID
classification math.NAcs.NA MSC 74S0574S1074S1574S2074S25
keywords Stokes-DarcycouplingnonconformingfiniteelementCrouzeix-RaviartstabilizedmixedmethodBeavers-Joseph-Saffmanconditionapriorierroranalysisisotropicmesh
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper develops and analyzes a unified stabilized mixed finite element method for the coupled Stokes-Darcy problem in two and three dimensions. The key move is to modify the Darcy equations by adding a divergence term and then use one variant of the nonconforming Crouzeix-Raviart element in both the free-fluid and porous regions, with a stabilization term that penalizes jumps across element edges. The author proves that the discrete problem is well-posed and that, under smoothness assumptions on the exact solution, the combined velocity-pressure error is $O(h)$ in the natural discrete norm. This matters because existing unified methods either use conforming elements in both regions or restrict the Darcy space more strongly, while the scheme here is cheaper and relies on a space that matches the natural $H(\mathrm{div})$ regularity of Darcy flow.

What carries the argument

The two devices that carry the argument are the modified Darcy formulation and the enlarged Crouzeix-Raviart space $H_h$. The Darcy modification replaces the raw momentum equation by $(\mu K^{-1}u,v)_{\Omega_d} + (\mathrm{div}\,u,\mathrm{div}\,v)_{\Omega_d}$ plus pressure terms, which makes the bilinear form coercive over the relevant $H(\mathrm{div})$ space and allows the same element in both regions. The space $H_h$ defined in (28)-(29) requires average continuity of the entire velocity across edges in the Stokes region and of only the normal component across edges in the Darcy region and on the outer Darcy boundary; a stabilization term $J(u,v)$ with $h^{-1}$ weights penalizes the remaining jumps. The convergence proof combines the discrete inf-sup condition with interpolation estimates for the four face-integral consistency errors.

What would settle it

Compute the interpolation operator $r_h$ defined by (49)-(50) on a single isotropically refined mesh in two dimensions, using a Darcy-region velocity that has a nonzero tangential average jump across an interior edge; if $\|r_h v\|_h$ grows faster than a constant times $\|v\|_{1,d}$ as the mesh is refined, Lemma 3.2 is false and Theorem 3.4 is unsupported. Alternatively, run the Section 4 numerical test with a Darcy velocity in $H(\mathrm{div})$ but not $H^1$ to see whether the claimed $O(h)$ rate collapses.

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Extended reading notes

Core claim

The central claim is Theorem 3.4: if the exact velocity is $H^2$ in each subdomain and the pressures are $H^1$, then the finite element solution satisfies $\|u-u_h\|_h + \|p-p_h\| \leq C h (|u|_{2,s}+|u|_{2,d}+|p|_{1,s}+|p|_{1,d})$, with $C$ independent of the mesh size. The method achieves this by modifying the Darcy problem with an added $\mathrm{div}$-$\mathrm{div}$ term and by using a Crouzeix-Raviart space that enforces zero mean jump of the full velocity on Stokes-side edges but only zero mean jump of the normal component on Darcy-side edges, which is the natural continuity for $H(\mathrm{div})$ velocity fields. The discrete inf-sup condition is proved via a Fortin argument, and the error analysis handles the resulting nonconformity through explicit face-integral estimates.

Load-bearing premise

The convergence proof assumes that the interpolation operator onto the enlarged space $H_h$ is bounded in the discrete norm and that the quoted approximation estimates hold, but these properties were proven for a smaller space in a cited paper and are not verified for the normal-continuity-only Darcy space actually used.

Editorial extensions

If this is right

  • The discrete problem (31) is well-posed for any regular, interface-conforming triangulation, yielding a unique velocity-pressure pair in $H_h\times Q_h$.
  • For smooth solutions the method reaches the optimal first-order convergence rate in the discrete norm, so halving the mesh size halves the combined velocity-pressure error.
  • Because the same nonconforming element is used in both regions and the pressure is piecewise constant, the implementation is simpler than methods that pair different stable elements in the Stokes and Darcy domains.
  • The stabilization term that penalizes element-edge jumps is essential to the coercivity proof, so the scheme avoids the need for separate inf-sup stable pairs in each subdomain.
  • The numerical experiment on a two-dimensional model problem reproduces the predicted first-order convergence rates for both velocity and pressure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the missing interpolation bound for $H_h$ can be established directly rather than quoted from a smaller space, the same scheme would be fully self-contained and likely extend to anisotropic meshes, since jump-penalty terms similar to discontinuous Galerkin methods often tolerate anisotropy.
  • The added $\mathrm{div}$-$\mathrm{div}$ term in the Darcy modification is a natural stabilization that vanishes on exactly divergence-free Darcy velocities; this suggests the method may transfer to a Brinkman-type model where the same term acts as a consistent penalty.
  • A test with Darcy velocity in $H(\mathrm{div})$ but not $H^1$ would reveal whether the $O(h)$ rate depends on the stronger smoothness assumption or whether the normal-only continuity already captures the true regularity of the solution.
  • The face-integral estimates in Lemma 3.5 are written for isotropic meshes; porting them to anisotropic meshes would require a trace inequality with explicit dependence on element aspect ratios, which could be the next step toward a unified anisotropic analysis.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes a unified stabilized mixed finite element method for the coupled Stokes–Darcy problem. The Darcy equation is modified by adding a div-div term, and both subproblems are discretized with a variant of the nonconforming Crouzeix–Raviart element in a single velocity space, with full-vector jump vanishing only in the Stokes region and normal-component jump vanishing in the Darcy region and on its boundary, together with a jump stabilization term. The paper claims well-posedness of the discrete problem, a discrete inf-sup condition, and an optimal first-order a priori error estimate under H^2/H^1 regularity, and reports one numerical experiment in two dimensions that shows the expected convergence behavior.

Significance. If the proof were completed, the result would be genuinely useful: it would justify a cheap, unified nonconforming discretization in which the Darcy velocity space only approximates H(div), which is more natural than enforcing full H^1 continuity in the porous medium. The scheme itself and the stabilization are reasonable, and the numerical experiment is consistent with the claimed rate. However, the central convergence argument depends on an interpolation operator that is not shown to map into the actual discrete space and is deferred to a reference that treats a smaller space. As written, the discrete inf-sup condition and the convergence theorem are not established. The contribution is promising, but the missing proof is load-bearing and requires a substantive repair.

major comments (3)
  1. [Section 3.1, Lemma 3.2 and equations (28)-(29), (49)-(50)] The interpolation operator r_h defined by (49)-(50) is not shown to map W into the actual space H_h. H_h in (28)-(29) requires full-vector jump moments to vanish on E_h(Ω_s^+) and normal-component jump moments to vanish on E_h(Ω_d) ∪ E_h(∂Ω_d), whereas (49)-(50) only fix edge moments on E_h(Ω_s) and E_h(Ω_d). In particular, the boundary-edge conditions on Γ_s, Γ_d, and Γ_I are not imposed by (49)-(50): for v∈W with v=0 on Γ_s and v·n_d=0 on Γ_d, there is no reason that r_h v satisfies the corresponding discrete conditions. Since Lemma 3.2 is proved only by saying "similar to [30]" and the space in [30] is smaller, the boundedness of r_h in the discrete norm is unproved for the enlarged H_h. This lemma underpins both the Fortin argument and the approximation estimate in Lemma 3.4, so the O(h) claim in Theorem 3.4 is unsupported as written.
  2. [Section 3.1, Theorem 3.2 (Fortin argument)] The Fortin proof integrates by parts over all elements and then restricts the resulting jump sum to E_h(Ω_s^+) ∪ E_h(Ω_d), omitting the edges in E_h(∂Ω_d). Because r_h v is not known to lie in H_h, the boundary terms on Γ_d and Γ_I are not zero, so the identity b_h(r_h v, q_h) = b_h(v, q_h) does not follow from (49)-(50). For v∈[H^1_0(Ω)]^N, the term Σ_{E⊂∂Ω_d} ∫_E q_h n_E·(r_h v) is generally nonzero unless additional conditions are imposed on r_h. Thus the discrete inf-sup condition (52) is not established.
  3. [Section 3.1, Theorem 3.1 and equations (43)-(47)] The coercivity proof contains some nontrivial steps that are only sketched: the transition from (42) to (43) and then to (44) relies on a discrete Korn inequality for a space with mixed jump conditions, but the precise hypotheses under which [5] applies to H_h are not stated. In addition, the inequalities in (45)-(47) use the symbol ">" where a lower bound with a constant is intended. These issues are repairable, but they add to the impression that the well-posedness proof is not fully self-contained.
minor comments (4)
  1. [Section 3.2, Lemma 3.5] The expressions ([v_h·n_E, p_s]_E)_E and ([v_h·n_E, p_d]_E)_E are typographically confusing; they should be written as ([v_h·n_E]_E, p_s)_E and ([v_h·n_E]_E, p_d)_E, respectively.
  2. [Section 3.2, Theorem 3.4] The final bound in (72) contains the typo |u2,d; it should read |u|_{2,d}. The word "satisfies" is also misspelled in the theorem statement.
  3. [Section 4, numerical experiments] Only one numerical test is reported, and the observed convergence orders are not extracted or tabulated. Reporting the empirical orders for ‖u−u_h‖_h and ‖p−p_h‖ would make the numerical confirmation of the O(h) claim more convincing.
  4. [Section 3.1, notation] The decomposition E_h = E_h(Ω_s^+) ∪ E_h(Ω_d) ∪ E_h(∂Ω_d) in (24) should be stated with a short explanation that E_h(Ω_s^+) includes Γ_s, while E_h(∂Ω_d) includes Γ_d and Γ_I; this would help the reader see exactly which jump terms are penalized.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the discrete error analysis is derived in-paper; deferred interpolation lemmas create a correctness gap, not circularity.

full rationale

The central claim, Theorem 3.4, is an a priori error bound obtained through a standard Strang-lemma chain. Coercivity (Theorem 3.1), the Fortin-based discrete inf-sup condition (Theorem 3.2), and the consistency term estimates (Lemma 3.5) are argued in the paper; no fitted parameter is renamed as a prediction, and no equation defines the predicted convergence rate as an input. Theorem 2.1 is attributed to the author's earlier work [27], but it is a standard mixed well-posedness result whose proof ingredients (continuity, inf-sup, coercivity plus classical mixed theory) are indicated in the text, and it does not by construction imply the discrete O(h) bound. The numerical experiment manufactures an exact solution and solves for the corresponding right-hand sides, so the observed rates are independent checks rather than fits. The main weakness flagged in the derivation—Lemma 3.2 and Lemma 3.4 are deferred to [30] while the present H_h is larger, requiring only normal continuity on Darcy edges—is a verification and correctness gap, not a circular reduction: nothing in the paper equates the target error with the assumed interpolation bound by definition. Accordingly, per the hard rules, this is not circularity.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The proof rests on standard mixed FEM theory, a discrete Korn inequality, and interpolation estimates, several of which are cited rather than demonstrated. The most important unproved ingredient is Lemma 3.2, which is tailored to the paper's larger CR space. No entirely new physical or mathematical entities are introduced.

free parameters (1)
  • Stabilization coefficient = (1+2\mu) in J_{\Omega_s^+}; 1 in J_{\Omega_d} and J_{\partial\Omega_d}
    These weights appear in the definition of the bilinear form J (equation (32)) and are chosen by hand. The analysis only requires them to be positive, so they are not fitted to data, but the specific values are arbitrary design parameters.
assumptions (6)
  • standard math Continuous well-posedness of the modified variational problem (21) (Theorem 2.2).
    Cited to [27] without proof; the present author is a co-author of [27], so this is a self-cited premise.
  • standard math Boundedness of the Crouzeix-Raviart interpolant r_h in the discrete norm (Lemma 3.2).
    Proof deferred to [30] with 'similar to [30]', but the space H_h here has weaker continuity conditions, so the lemma is not actually demonstrated.
  • standard math Discrete Korn inequality for piecewise H^1 functions, applied in Theorem 3.1.
    Referenced to [5] (Brenner); the specific form with the global curl term \varphi(v_h) in place of the full jump term is not standard and is not proved.
  • domain assumption Shape-regular conforming triangulations aligned with the Stokes/Darcy subdomains.
    Stated in Section 3.1; the error analysis and the numerical test use isotropic (shape-regular) meshes.
  • domain assumption The permeability tensor K is symmetric, uniformly positive definite with bounds K_* and K^*; the source g has zero mean.
    Standard assumptions on the model stated in Section 2.1.
  • standard math Existence of a Fortin operator for the pair (H_h, Q_h) and standard interpolation error estimates.
    Used in Theorem 3.2 and Lemma 3.4; these are standard results but not fully proved in the paper.

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Pith. "Pith review of A new unified stabilized mixed finite element method of the Stokes-Darcy coupled problem: Isotropic discretization." pith.science (2026). https://pith.science/paper/7UBUUBXD

@misc{pith2026190801892,
  author       = {Pith},
  title        = {Pith review of: A new unified stabilized mixed finite element method of the Stokes-Darcy coupled problem: Isotropic discretization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7UBUUBXD}},
  note         = {Machine review of arXiv:1908.01892}
}
abstract

In this paper we develop an a priori error analysis of a new unified mixed finite element method for the coupling of fluid flow with porous media flow in $\mathbb{R}^N$, $N\in\{2,3\}$ on isotropic meshes. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The approach utilizes a modification of the Darcy problem which allows us to apply a variant nonconforming Crouzeix-Raviart finite element to the whole coupled Stokes-Darcy problem. The well-posedness of the finite element scheme and its convergence analysis are derived. Finally, the numerical experiments are presented, which confirm the excellent stability and accuracy of our method.

Figures

Figures reproduced from arXiv: 1908.01892 by the authors.

Figure 2
Figure 2. A sketch of the geometry of the problem (case ∂Ωd = ΓI .) For any function v defined in Ω, since its restriction to Ωs or to Ωd could play a different mathematical roles (for instance their traces on ΓI ), we will set vs = v|Ωs and vd = v|Ωd . In Ω, we denote by u the fluid velocity and by p the pressure. The motion of the fluid in Ωs is described by the Stokes equations    −2µ div D(u) + ∇p = f in Ωs, div u = g … view at source ↗
Figure 6
Figure 6. P 1 -nonconforming finite element T in 2d. For i ∈ {0, · · · , N}, we set: σi(p) := 1 |Ei | Z Ei p, ∀p ∈ P 1 (T), where Ei ∈ E(T) (25) The triplet {T, P 1 (T), Σ} with Σ = {σi}0 6i 6N is finite element [10, Page 83]. The local basis functions are defined by: ψi(T) = 1 − Nλi(T), i ∈ {0, . . . , N}, (26) [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 10
Figure 10. The domain Ω in 2d. We consider the application φ : (x, y) ∈ R 2 7−→ φ(x, y) = x 2 (x − 1)3 y 2 (y − 1)2 ∈ R on the square Ω =]0, 1[2∪]1, 2[2 . In Ω, we define u = (u1, u2) = curl φ  − ∂φ ∂y , ∂φ ∂x  and we obtain: u1(x, y) := −2(−1 + x) 3x 2 (−1 + y)y(−1 + 2y) (73) u2(x, y) := (−1 + x) 2x(−2 + 5x)(−1 + y) 2 y 2 (74) We choose quadratic pressure p ∈ L 2 (Ω) by p(x, y) = x 2 − 2xy + y 2 2 − 1. (75) [PITH_FULL_IMAG… view at source ↗
Figures from the paper (10 more)
Figure 12
Figure 12. Figure 12: Example of anisotropic mesh in 2d [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: Isotropic mesh on coupled domain Ω ⊂ R2 [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: Error for the velocity k u − uh kh in Ωs ( log/log plot) [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 16
Figure 16. Figure 16: Error for the velocity k u − uh kh in Ωd (log/log plot) [PITH_FULL_IMAGE:figures/full_fig_p021_16.png]
Figure 18
Figure 18. Figure 18: The isovalue of the first velocity component u1 in Ωs [PITH_FULL_IMAGE:figures/full_fig_p021_18.png]
Figure 20
Figure 20. Figure 20: The isovalue of the first velocity component u1 in Ωd [PITH_FULL_IMAGE:figures/full_fig_p021_20.png]
Figure 22
Figure 22. Figure 22: Component u1 in Ω [PITH_FULL_IMAGE:figures/full_fig_p022_22.png]
Figure 24
Figure 24. Figure 24: Pressure p in Ω [PITH_FULL_IMAGE:figures/full_fig_p022_24.png]
Figure 25
Figure 25. Figure 25: Right-hand term f1 in Ωs [PITH_FULL_IMAGE:figures/full_fig_p022_25.png]
Figure 27
Figure 27. Figure 27: Right-hand term k1 in Ωd [PITH_FULL_IMAGE:figures/full_fig_p023_27.png]

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Works this paper leans on

34 extracted references · 31 canonical work pages

  1. [30]

    Rui and R

    H. Rui and R. Zhang. A unified stabilized mixed finite element method for coupling Stokes and Darcy flows. Comput. Methods Appl. Mech. Engrg. , 198:2692–2699, 2009

    work page 2009

  2. [5]

    S. Brenner. Korn’s inequalities for piecewise H 1 vector fields. Math. Comput., 73:1067–1087, 2003

    work page 2003

  3. [1]

    Arbogast and D

    T. Arbogast and D. Brunson. A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium. Computational Geosciences, 11:207–218, 2007

    work page 2007

  4. [2]

    M. G. Armentano and M. L. Stockdale. A unified mixed finite element approximations of the Stokes-Darcy cou- pled problem. Computers and Mathematics with Applications , https://doi.org/10.1016/j.camwa.2018.12.032, 2018

    doi:10.1016/j.camwa.2018.12.032 2018

  5. [3]

    Babuˇ ska and G

    I. Babuˇ ska and G. Gatica. A residual-based a posteriori error estimator for the Stokes-Darcy coupled problem. SIAM J. Numer. Anal. , 48:498–523, 2010

    2010

  6. [4]

    Beavers and D

    G. Beavers and D. Joseph. Boundary conditions at a naturally permeable wall. J. Fluid Mech. , 30:197–207, 1967

    work page 1967

  7. [6]

    Chen and Y

    W. Chen and Y. Wang. A posteriori error estimate for H(div) conforming mixed finite element for the coupled Darcy-Stokes system. Journal of Computational and Applied Mathematics , 255:502–516, 2014

    work page 2014

  8. [7]

    Crouzeix and P

    M. Crouzeix and P. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. Rev. Fran¸ caise Automat. Informat. Recherche Op´ erationnelle s´ er. Rouge, 7:33–75, 1973

    work page 1973

Show all 34 references
  1. [8]

    Cui and N

    M. Cui and N. Yan. A posteriori error estimate for the Stokes-Darcy system. Math. Meth. Appl. Sci. , 34:1050– 1064, 2011

    2011

  2. [9]

    Discacciati and A

    M. Discacciati and A. Quarteroni. Navier-Stokes/Darcy coupling: Modeling, analysis, and numerical approxi- mation. Rev. Math. Comput. , 22:315–426, 2009

    2009

  3. [10]

    A. Ern. Aide-m´ emoire el´ ements finis.Dunod, Paris, ISBN 2 10 007303 6 , 2005

    2005

  4. [11]

    Frederic and P

    H. Frederic and P. Olivier. Freefem++. http://www.freefem.org

  5. [12]

    Galvis and M

    J. Galvis and M. Sarkis. Nonconforming mortar discretization analysis for the coupling Stokes-Darcy equations. Electronic. Trans. Numer. Anal., 26:350–384, 2007

    2007

  6. [13]

    Gatica, R

    G. Gatica, R. Oyarz` ua, and F.-J. Sayas. A residual-based a posteriori error estimator for a fully-mixed formu- lation of the Stokes-Darcy coupled problem. Comput. Methods Appl. Mech. Engrg. , 200:1877–1891, 2011

    2011

  7. [14]

    G. N. Gatica, S. Meddahi, and R. Oyarz` ua. A conforming mixed finite element method for the coupling of fluid flow with porous media flow. IMA J. Numer. Anal. , 29:86–108, 2009

    2009

  8. [15]

    Gatica, R

    G.-N. Gatica, R. Oyarz` ua, and F.-J. Sayas. Convergence of a family of galerkin discretizations for the Stokes- Darcy coupled proplem

  9. [16]

    Girault and P.-A

    V. Girault and P.-A. Raviart. Finite element methods for Navier-Stokes equations, Theory and algorithms , volume 5 of Springer, Berlin. In Computational Mathematics, 1986

    1986

  10. [17]

    F. Hecht. The mesh adapting software: bamg. INRIA report.http://www- c.inria.fr/gamma/cdrom/www/bamg/eng.htm, 1998. 24 HOU ´EDANOU KOFFI WILFRID (A)

    1998

  11. [18]

    K. W. Hou´ edanou, J. Adetola, and B. Ahounou. Residual-based a posteriori error estimates for a conform- ing finite element discretization of the Navier-Stokes/Darcy coupled problem. Journal of Pure and Applied Mathematics : Advances and Applications , 18(1):37–73, 2017

    2017

  12. [19]

    K. W. Hou´ edanou and B. Ahounou. A posteriori error estimation for the Stokes-Darcy coupled problem on anisotropic discretization. Math. Meth. Appl. Sci. , 2016. http://dx.doi.org/10.1002/mma.4261 (in press)

    2016 doi

  13. [20]

    J¨ ager and A

    W. J¨ ager and A. Mikeli´ c. On the boundary conditions of the contact interface between a porous medium and a free fluid. Ann. Scuola Norm. Sup. Oisa Cl. Sci. , 23:403–465, 1996

    1996

  14. [21]

    J¨ ager and A

    W. J¨ ager and A. Mikeli´ c. On the interface boundary condition of Beavers, Joseph and Saffman.SIAM Journal on Applied Mathematics , 60:1111–1127, 2000

    2000

  15. [22]

    J¨ ager and A

    W. J¨ ager and A. Mikeli´ c. On the interface boundary condition of Beavers, Joseph and Saffman.SIAM J. Appl. Math., 60:1111–1127, 2000

    2000

  16. [23]

    J¨ ager, A

    W. J¨ ager, A. Mikeli´ c, and N. Neuss. Asymptotic analysis of the laminar visous flow over a porous bed.SIAM J. Sci. Comput. , 22:2006–2028, 2001

    2006

  17. [24]

    R. Li, J. Li, Z. Chen, and Y. Gao. A stabilized finite element method based on two local gauss integrations for a coupled Stokes-Darcy problem. http://dx.doi.org/10.1016/j.cam.2015.06.014, 2015

    2015 doi

  18. [25]

    Mardal, X

    K.-A. Mardal, X. Tai, and R. Winther. A robust finite element method for darcy-stokes flow. SIAM Journal on Numerical Analysis, 40:1605–1631, 2002

    2002

  19. [26]

    Mu and J

    M. Mu and J. Xu. A two-grid method of a mixed Stokes-Darcy model for coupling fluid flow with porous media flow. SIAM Journal on Numerical Analysis , 45:1801–1813, 2007

    2007

  20. [27]

    Nicaise, B

    S. Nicaise, B. Ahounou, and W. Hou´ edanou. A residual-based posteriori error estimates for a nonconforming finite element discretization of the Stokes-Darcy coupled problem: Isotropic discretization. Afr. Mat., African Mathematical Union and Springer-Verlag Berlin Heidelberg: ...

    2016

  21. [28]

    Payne and B

    L. Payne and B. Straughan. Analysis of the boundary condition at the interface between a viscous fluid and a porous medium and related modeling questions. J. Math. Pures Appl. , 77:317–354, 1998

    1998

  22. [29]

    Rivi` ere and I

    B. Rivi` ere and I. Yotov. Locally conservative coupling of Stokes and Darcy flows. SIAM J. Numer. Anal. , 42:1959–1977, 2005

    1959

  23. [31]

    P. Saffman. On the boundary condition at the interface of a porous medium. Stud. Appl. Math., 1:93–101, 1971

    1971

  24. [32]

    Vassilev and I

    D. Vassilev and I. Yotov. Coupling Stokes-Darcy flow with transport. SIAM J. Sci. Comput. , 5:3661–3684, 2009

    2009

  25. [33]

    L. J. William, S. Friedhelm, and Y. Ivan. Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. , 40(6):2195–2218 (2003), 2002

    2003

  26. [34]

    J. Yu, M. A. A. Mahbub, F. Shi, and H. Zheng. Stabilized finite element method for the stationary mixed Stokes- Darcy problem. Advances in Diffference Equations, https:// doi.org/10.1186/s13662-018-1809-2346, 2018. E-mail address: a) khouedanou@yahoo.fr D´epartement de Math´emat...

    2018 doi

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