Vorticity‐pressure formulations for the Brinkman‐Darcy coupled problem

Anaya, Verónica; Mora, David; Reales, Carlos; Ruiz–Baier, Ricardo

doi:10.1002/num.22312

Cited by 6 publications

(3 citation statements)

References 29 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%

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%

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 begin the solvability analysis with the following result, whose proof is a direct consequence of Alvarez et al (2016b, Theorem 3.2). Let us remark that similar vorticity-based formulations for Brinkman-Darcy equations can be analysed using a different approach, as done recently in Anaya et al (2019). Lemma 3.1 For each φ ∈ H 1 Γ 0 (Ω) problem (3.4) has a unique solution ( u, p) ∈ H × Q 0 .…”

Section: Well Posedness Of the Uncoupled Problemmentioning

confidence: 99%

A mixed-primal finite element method for the coupling of Brinkman–Darcy flow and nonlinear transport

Alvarez

Gatica

Ruiz–Baier

2020

IMA Journal of Numerical Analysis

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This paper is devoted to the mathematical and numerical analysis of a model describing the interfacial flow-transport interaction in a porous-fluidic domain. The medium consists of a highly permeable material, where the flow of an incompressible viscous fluid is governed by Brinkman equations (written in terms of vorticity, velocity and pressure), and a porous medium where Darcy’s law describes fluid motion using filtration velocity and pressure. Gravity and the local fluctuations of a scalar field (representing for instance, the solids volume fraction or the concentration of a contaminant) are the main drivers of the fluid patterns on the whole domain, and the Brinkman-Darcy equations are coupled to a nonlinear transport equation accounting for mass balance of the scalar concentration. We introduce a mixed-primal variational formulation of the problem and establish existence and uniqueness of solution using fixed-point arguments and small-data assumptions. A family of Galerkin discretizations that produce divergence-free discrete velocities is also presented and analysed using similar tools to those employed in the continuous problem. Convergence of the resulting mixed-primal finite element method is proven, and some numerical examples confirming the theoretical error bounds and illustrating the performance of the proposed discrete scheme are reported.

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“…For the specific application of interfacial flow in the eye, the radial symmetry of the domain and of the flow conditions could be better represented using axisymmetric formulations as in [7,22,39]. Then, the domain as well as the expected flow properties are all symmetric with respect to the axis of symmetry Γ axisymm .…”

Section: Axisymmetric Casementioning

confidence: 99%

The Biot-Stokes coupling using total pressure: formulation, analysis and application to interfacial flow in the eye

Ruiz-Baier,

Taffetani,

Westermeyer

et al. 2021

Preprint

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We consider a multiphysics model for the flow of Newtonian fluid coupled with Biot consolidation equations through an interface, and incorporating total pressure as an unknown in the poroelastic region. A new mixed-primal finite element scheme is proposed solving for the pairs fluid velocitypressure and displacement -total poroelastic pressure using Stokes-stable elements, and where the formulation does not require Lagrange multipliers to set up the usual transmission conditions on the interface. The stability and well-posedness of the continuous and semi-discrete problems are analysed in detail. Our numerical study is framed in the context of different interfacial flow regimes in Cartesian and axisymmetric coordinates that could eventually help describe early morphologic changes associated with glaucoma development in canine species.

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Vorticity‐pressure formulations for the Brinkman‐Darcy coupled problem

Cited by 6 publications

References 29 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

A mixed-primal finite element method for the coupling of Brinkman–Darcy flow and nonlinear transport

The Biot-Stokes coupling using total pressure: formulation, analysis and application to interfacial flow in the eye

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