A vorticity-based fully-mixed formulation for the 3D Brinkman–Darcy problem

Alvarez, Mario; Gatica, Gabriel N.; Ruiz–Baier, Ricardo

doi:10.1016/j.cma.2016.04.017

Cited by 23 publications

(25 citation statements)

References 28 publications

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“…A smooth interface exists between the Darcy and Brinkman subdomains, where the Brinkman part is on top (see related test cases in [1,4,10,14]). For this problem we assume a uniform current flow on the x 1 −direction and the presence of gravity, so f B = f D = (0.25, 0, −0.1) T .…”

Section: Numerical Resultsmentioning

confidence: 99%

“…According to the previous discussion, and as a natural continuation of the a priori error analysis developed in [1], our goal in the present paper is to provide a reliable and efficient residual-based a posteriori error estimator for the finite element method introduced and analyzed in that reference. In this way, we aim to improve the accuracy of the discrete scheme from [1] in different scenarios, including presence of singularities or high gradients of the solution.…”

Section: Introductionmentioning

confidence: 99%

“…We have recently introduced a mixed finite element method to numerically approximate the flow patterns of a viscous fluid within a highly permeable medium described by Brinkman equations, and its interaction with pure porous media flow under Darcy's law [1]. There, the system is formulated in terms of velocity and pressure of the nonviscous flow, together with vorticity, velocity and pressure of the Brinkman region.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, we refer to [6,[9][10][11]14,16,27], which are focused on Stokes-Darcy and Navier-Stokes/Darcy couplings, and where the interface conditions are treated in different ways, from both mathematical and numerical perspective. For instance, in [6,14,16,27], Beavers-Joseph-Saffman-type conditions are considered on the interface, whereas in [10,11], similar transmission conditions to those employed in [1] are assumed. Also, an interesting feature of the proof of reliability in [6], which differs from the approaches in the other works, is the utilisation of intermediate inf-sup inequalities that are obtained along the proof of the global inf-sup condition.…”

Section: Introductionmentioning

confidence: 99%

“…In this way, we aim to improve the accuracy of the discrete scheme from [1] in different scenarios, including presence of singularities or high gradients of the solution. Indeed, in contrast with the methodology developed in [10,11], and following the approaches in [16,27], we highlight that the derivation of our error estimator is based on a global inf-sup condition in combination with suitable Helmholtz decompositions adapted from [16,31], and local approximation properties of Clément, Raviart-Thomas, and Clément-type Nédélec interpolators.…”

Section: Introductionmentioning

confidence: 99%

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A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem

Alvarez

Gatica

Ruiz–Baier

2017

Calcolo

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We develop the a posteriori error analysis for a mixed finite element method applied to the coupling of Brinkman and Darcy equations in 3D, modelling the interaction of viscous and non-viscous flow effects across a given interface. The system is formulated in terms of velocity and pressure within the Darcy subdomain, together with vorticity, velocity and pressure of the fluid in the Brinkman region, and a Lagrange multiplier enforcing pressure continuity across the interface. The solvability of a fullymixed formulation along with a priori error bounds for a finite element method have been recently established in Álvarez et al. ( Comput Methods Appl Mech Eng 307:68-95, 2016). Here we derive a residual-based a posteriori error estimator for such a scheme, and prove its reliability exploiting a global inf-sup condition in combination with suitable Helmholtz decompositions, and interpolation properties of Clément and Raviart-Thomas operators. The estimator is also shown to be efficient, following a localisation strategy and appropriate inverse inequalities. We present numerical tests to confirm the features of the estimator and to illustrate the performance of the method in academic and application-oriented problems.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem

Alvarez

Gatica

Ruiz–Baier

2017

Calcolo

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Vorticity‐pressure formulations for the Brinkman‐Darcy coupled problem

Anaya

Mora

Reales

et al. 2018

Numerical Methods Partial

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We introduce a new variational formulation for the Brinkman-Darcy equations formulated in terms of the scaled Brinkman vorticity and the global pressure. The velocities in each subdomain are fully decoupled through the momentum equations, and can be later recovered from the principal unknowns. A new finite element method is also proposed, consisting in equal-order Nédélec and piecewise continuous elements, for vorticity and pressure, respectively. The error analysis for the scheme is carried out in the natural norms, with bounds independent of the fluid viscosity. An adequate modification of the formulation and analysis permits us to specify the presentation to the case of axisymmetric configurations. We provide a set of numerical examples illustrating the robustness, accuracy, and efficiency of the proposed discretization. KEYWORDSBrinkman-Darcy coupling, error analysis, finite element methods, interface problems, vorticity-pressure formulation 1 Numer Methods Partial Differential Eq. 2019;35:528-544. wileyonlinelibrary.com/journal/num

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A priori and a posteriori error analysis of a mixed scheme for the Brinkman problem

et al. 2015

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This paper deals with the analysis of new mixed finite element methods for the Brinkman equations formulated in terms of velocity, vorticity and pressure. Employing the Babuška-Brezzi theory, it is proved that the resulting continuous and discrete variational formulations are well-posed. In particular, we show that RaviartThomas elements of order k ≥ 0 for the approximation of the velocity field, piecewise continuous polynomials of degree k + 1 for the vorticity, and piecewise polynomials of degree k for the pressure, yield unique solvability of the discrete problem. On the other hand, we also show that families of finite elements based on Brezzi-DouglasMarini elements of order k +1 for the approximation of velocity, piecewise continuous polynomials of degree k +2 for the vorticity, and piecewise polynomials of degree k for the pressure ensure the well-posedness of the associated Galerkin scheme. We note that these families provide exactly divergence-free approximations of the velocity field. We establish a priori error estimates in the natural norms with constants independent B Ricardo Ruiz-Baier

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A vorticity-based fully-mixed formulation for the 3D Brinkman–Darcy problem

Cited by 23 publications

References 28 publications

A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem

A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem

Vorticity‐pressure formulations for the Brinkman‐Darcy coupled problem

A priori and a posteriori error analysis of a mixed scheme for the Brinkman problem

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