Stabilized mixed approximation of axisymmetric Brinkman flows

“…The use of Stokes-tailored is analyzed in [34], where a new nonconforming element is constructed. In addition to the aforementioned works, one can also refer to [10,11,39,4,8,26,32,22,24,38,2,1,6,28,3,23] and the references therein for a glance at uniformly stable methods for the Brinkman problem. Recently, some novel methods have been proposed to solve the Brinkman problem on general quadrilateral and polygonal meshes, which is tricky and usually requires special treatment in order to deliver robust results with respect to the rough grids.…”

Section: Introductionmentioning

confidence: 99%

A new staggered DG method for the Brinkman problem robust in the Darcy and Stokes limits

Zhao

Chung

Lam

2020

Computer Methods in Applied Mechanics and Engineering

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In this paper we propose a novel staggered discontinuous Galerkin method for the Brinkman problem on general quadrilateral and polygonal meshes. The proposed method is robust in the Stokes and Darcy limits, in addition, hanging nodes can be automatically incorporated in the construction of the method, which are desirable features in practical applications. There are three unknowns involved in our formulation, namely velocity gradient, velocity and pressure. Unlike the original staggered DG formulation proposed for the Stokes equations in [30], we relax the tangential continuity of velocity and enforce different staggered continuity properties for the three unknowns, which is tailored to yield an optimal L 2 error estimates for velocity gradient, velocity and pressure independent of the viscosity coefficient. Moreover, by choosing suitable projection, superconvergence can be proved for L 2 error of velocity. Finally, several numerical results illustrating the good performances of the proposed method and confirming the theoretical findings are presented.

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“…A variational formulation for system Equations – can be derived as in Section 2. In particular, we repeat the arguments in Equations – together with Lemmas 1.2 and 1.3 from Anaya and coworkers , to obtain the following variational formulation: Find ( ω B , p ) ∈ Z a × Q a such that

{script𝒜}^{normala} (((),, ω_{normalB}, p), ((),, θ, q)) = {normalℱ}^{normala} (θ, q) \forall (θ, q) \in {boldZ}^{normala} \times {normalQ}^{normala},

where the associated functional spaces are

{boldZ}^{normala} ≔ \{φ \in {\overset{H}{true}}_{1}^{1} (Ω_{B}^{a}) : φ = 0 on Σ^{normala}\}, {normalQ}^{normala} ≔ {normalH}_{1}^{1} (Ω^{a}) \cap {normalL}_{1, 0}^{2} (Ω^{a}),

and the bilinear form

{script𝒜}^{normala} : ({boldZ}^{normala} \times {normalQ}^{normala}) \times ({boldZ}^{normala} \times {normalQ}^{normala}) \to ℝ

and linear functional ℱ a : Z a × Q a → ℝ are now specified as

\begin{array}{l} 𝒜^{a} ((),, (ω_{B}, p), (θ, q)) ≔ & \int_{Ω_{normalB}^{normala}} ω_{B} θ r normald r normald z + \int_{Ω_{normalB}^{normala}} \end{array}

…”

Section: Reduction To the Axisymmetric Casementioning

confidence: 99%

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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Stabilized mixed approximation of axisymmetric Brinkman flows

Cited by 22 publications

References 45 publications

Discontinuous finite volume element discretization for coupled flow–transport problems arising in models of sedimentation

Discontinuous finite volume element discretization for coupled flow–transport problems arising in models of sedimentation

A new staggered DG method for the Brinkman problem robust in the Darcy and Stokes limits

Vorticity‐pressure formulations for the Brinkman‐Darcy coupled problem

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