Velocity-vorticity-pressure formulation for the Oseen problem with variable viscosity

Anaya, Verónica; Caraballo, Rubén; Gómez-Vargas, Bryan; Mora, David; Ruiz–Baier, Ricardo

doi:10.1007/s10092-021-00433-6

Cited by 7 publications

(5 citation statements)

References 44 publications

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“…where the unknowns are the velocity field u and the scalar pressure p. In addition, the constant κ > 0 is a length scale parameter characterizing the elasticity of the fluid, ν > 0 is the Brinkman coefficient (or the effective viscosity), D > 0 is the Darcy coefficient, F > 0 is the Forchheimer coefficient, and ρ ∈ [3,4] is a given number.…”

Section: The Model Problem and Its Velocity-vorticity-pressure Formul...mentioning

confidence: 99%

“…[6]). We observe that similarly to [3] and [2], we can also consider discontinuous piecewise polynomials spaces for the vorticity, that is,…”

Section: Existence and Uniqueness Of A Solutionmentioning

confidence: 99%

“…The purpose of the present work is to develop and analyze a new vorticity-based mixed formulation of the KVBF problem and to study a suitable conforming numerical discretization. To that end, unlike previous KVBF works and motivated by [4], [3], and [2], we introduce the vorticity as an additional unknown besides the fluid velocity and pressure. In addition to the advantage of providing a direct, accurate, and smooth approximation of the vorticity, our approach gives optimal theoretical convergence rates without requiring any small data or quasi-uniformity assumptions on the mesh.…”

Section: Introductionmentioning

confidence: 99%

“…In addition to the advantage of providing a direct, accurate, and smooth approximation of the vorticity, our approach gives optimal theoretical convergence rates without requiring any small data or quasi-uniformity assumptions on the mesh. Furthermore, unlike [4], [3], or [2], our method does not require any augmentation process. It is also important to mention that another novelty and advantage of the present work is that it generalizes the model studied in [2] by including the nonlinear convective term and an additional time-derivative term, thus considering viscoelastic flows.…”

Section: Introductionmentioning

confidence: 99%

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A Banach space mixed formulation for the unsteady Brinkman–Forchheimer equations

Caucao

Yotov

2020

IMA Journal of Numerical Analysis

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We propose and analyse a mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Our approach is based on the introduction of a pseudostress tensor related to the velocity gradient and pressure, leading to a mixed formulation where the pseudostress tensor and the velocity are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators and a regularization technique. We then present well posedness and error analysis for semidiscrete continuous-in-time and fully discrete finite element approximations on simplicial grids with spatial discretization based on the Raviart–Thomas spaces of degree $k$ for the pseudostress tensor and discontinuous piecewise polynomial elements of degree $k$ for the velocity and backward Euler time discretization. We provide several numerical results to confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method for a range of model parameters.

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Section: The Model Problem and Its Velocity-vorticity-pressure Formul...mentioning

confidence: 99%

“…[6]). We observe that similarly to [3] and [2], we can also consider discontinuous piecewise polynomials spaces for the vorticity, that is,…”

Section: Existence and Uniqueness Of A Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Banach space mixed formulation for the unsteady Brinkman–Forchheimer equations

Caucao

Yotov

2020

IMA Journal of Numerical Analysis

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“…The formulation of viscous flow equations using vorticity, velocity and pressure has been used and analysed (in terms of solvability of the continuous and discrete formulations and deriving error estimates) extensively in, for example, [1, 2, 6–9, 14, 23, 27–28, 56]. Methods based on vorticity formulations are useful for visualization of rotational flows and they are convenient when dealing with rotation‐based boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Robust finite element methods and solvers for the Biot–Brinkman equations in vorticity form

Caraballo,

In,

Martín

et al. 2023

Numerical Methods Partial

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In this article, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence‐conforming filtration fluxes. The proposed method is based on the use of parameter‐weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in a variety of regimes, including the case of large Lamé parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification and testing of robustness of block‐diagonal preconditioners with respect to model parameters.

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On augmented finite element formulations for the Navier–Stokes equations with vorticity and variable viscosity

Anaya

Caraballo

Ruiz–Baier

et al. 2023

Computers & Mathematics with Applications

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Velocity-vorticity-pressure formulation for the Oseen problem with variable viscosity

Cited by 7 publications

References 44 publications

A Banach space mixed formulation for the unsteady Brinkman–Forchheimer equations

A Banach space mixed formulation for the unsteady Brinkman–Forchheimer equations

Robust finite element methods and solvers for the Biot–Brinkman equations in vorticity form

On augmented finite element formulations for the Navier–Stokes equations with vorticity and variable viscosity

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