Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem

Gatica, Gabriel N.; Oyarzúa, Ricardo; Sayas, Francisco–Javier

doi:10.1090/s0025-5718-2011-02466-x

Cited by 84 publications

(77 citation statements)

References 32 publications

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“…Then, there exists α > 0, such that (27) and div w S = 0 in Ω S . Integrating by parts, it is easy to see that…”

Section: Well-posedness Of the Oseen-darcy Problemmentioning

confidence: 99%

“…So far, several numerical methods have been developed to approximate the solution of the Stokes-Darcy coupled problem (see for instance [13,14,18,19,20,22,26,27,28,32,36,37,39,40,41,7]), most of them based on appropriate combinations of stable elements for both media. In this direction, the first theoretical results go back to [39] and [20].…”

Section: Introductionmentioning

confidence: 99%

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A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

Discacciati

Oyarzúa

2016

Numer. Math.

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Citation: DISCACCIATI, M. and OYARZUA, R., 2016. A conforming mixed finite element method for the Navier Stokes/Darcy coupled problem. Numerische Mathematik, 135(2), pp. 571 606. Abstract In this paper we develop the a priori analysis of a mixed finite element method for the coupling of fluid flow with porous media flow. Flows are governed by the Navier-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. We consider the standard mixed formulation in the Navier-Stokes domain and the dual-mixed one in the Darcy region, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The finite element subspaces defining the discrete formulation employ Bernardi-Raugel and RaviartThomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for the Lagrange multiplier. We show stability, convergence, and a priori error estimates for the associated Galerkin scheme. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are reported.

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“…Then, there exists α > 0, such that (27) and div w S = 0 in Ω S . Integrating by parts, it is easy to see that…”

Section: Well-posedness Of the Oseen-darcy Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

Discacciati

Oyarzúa

2016

Numer. Math.

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“…[5], [14], [20], [22], and the references therein). This study has been extended to solve important problems in engineering, such as the Stokes-Darcy coupled problem and transport problems (see, for instance, [1], [21], [24], [25]). …”

Section: Introductionmentioning

confidence: 99%

Analysis of an augmented mixed-FEM for the Navier-Stokes problem

Camaño¹,

Oyarzúa²,

Tierra

2016

Math. Comp.

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In this paper we propose and analyze a new augmented mixed finite element method for the Navier-Stokes problem. Our approach is based on the introduction of a "nonlinearpseudostress" tensor linking the pseudostress tensor with the convective term, which leads to a mixed formulation with the nonlinear-pseudostress tensor and the velocity as the main unknowns of the system. Further variables of interest, such as the fluid pressure, the fluid vorticity and the fluid velocity gradient, can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. The resulting mixed formulation is augmented by introducing Galerkin least-squares type terms arising from the constitutive and equilibrium equations of the Navier-Stokes equations and from the Dirichlet boundary condition, which are multiplied by stabilization parameters that are chosen in such a way that the resulting continuous formulation becomes well-posed. Then, the classical Banach's fixed point Theorem and Lax-Milgram's Lemma are applied to prove well-posedness of the continuous problem. Similarly, we establish well-posedness and the corresponding Cea's estimate of the associated Galerkin scheme considering any conforming finite element subspace for each unknown. In particular, the associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree k for the nonlinear-pseudostress tensor, and continuous piecewise polynomial elements of degree k + 1 for the velocity, which leads to an optimal convergent scheme. In addition, we provide two iterative methods to solve the corresponding nonlinear system of equations and analyze their convergence. Finally, several numerical results illustrating the good performance of the method are provided.

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“…In particular, the development of suitable numerical methods for solving the Stokes-Darcy and related coupled problems, including porous media with cracks, the incorporation of the Brinkman equation in the model, and linear as well as nonlinear behaviors, has become a very active research area during the last decade (see, e.g. [13], [23], [25], [26], [31], [32], [37], [48], [55] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

A mixed finite element method for Darcy’s equations with pressure dependent porosity

2015

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In this work we develop the a priori and a posteriori error analyses of a mixed finite element method for Darcy's equations with porosity depending exponentially on the pressure. A simple change of variable for this unknown allows to transform the original nonlinear problem into a linear one whose dual-mixed variational formulation falls into the frameworks of the generalized linear saddle point problems and the fixed point equations satisfied by an affine mapping. According to the latter, we are able to show the well-posedness of both the continuous and discrete schemes, as well as the associated Cea estimate, by simply applying a suitable combination of the classical Babuška-Brezzi theory and the Banach fixed point Theorem. In particular, given any integer k ≥ 0, the stability of the Galerkin scheme is guaranteed by employing Raviart-Thomas elements of order k for the velocity, piecewise polynomials of degree k for the pressure, and continuous piecewise polynomials of degree k+1 for an additional Lagrange multiplier given by the trace of the pressure on the Neumann boundary. Note that the two ways of writing the continuous formulation suggest accordingly two different methods for solving the discrete schemes. Next, we derive a reliable and efficient residualbased a posteriori error estimator for this problem. The global inf-sup condition satisfied by the continuous formulation, Helmholtz decompositions, and the local approximation properties of the Raviart-Thomas and Clément interpolation operators are the main tools for proving the reliability. In turn, inverse and discrete inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of both methods, confirming the aforementioned properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are reported.

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Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem

Cited by 84 publications

References 32 publications

A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

Analysis of an augmented mixed-FEM for the Navier-Stokes problem

A mixed finite element method for Darcy’s equations with pressure dependent porosity

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