A five-field augmented fully-mixed finite element method for the Navier–Stokes/Darcy coupled problem

Gatica, Gabriel N.; Oyarzúa, Ricardo; Valenzuela, Nathalie

doi:10.1016/j.camwa.2020.08.017

Cited by 6 publications

(3 citation statements)

References 26 publications

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“…The coupling of fluid flows with porous media flows involving the stationary incompressible Navier-Stokes equations with constant viscosity and the Darcy equations with a permeability given in terms of a uniformly elliptic matrix-valued function with L ∞ coefficients has been recently investigated in [27] (see also [26]).…”

Section: Introductionmentioning

confidence: 99%

Non-homogeneous Dirichlet-transmission problems for the anisotropic Stokes and Navier-Stokes systems in Lipschitz domains with transversal interfaces

2022

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This paper is build around the stationary anisotropic Stokes and Navier-Stokes systems with an $$L^\infty $$ L ∞ -tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices in $${\mathbb {R}}^{n\times n}$$ R n × n with zero matrix traces. We analyze, in $$L^2$$ L 2 -based Sobolev spaces, the non-homogeneous boundary value problems of Dirichlet-transmission type for the anisotropic Stokes and Navier-Stokes systems in a compressible framework in a bounded Lipschitz domain with a transversal Lipschitz interface in $${\mathbb {R}}^n$$ R n , $$n\ge 2$$ n ≥ 2 ($$n=2,3$$ n = 2 , 3 for the nonlinear problems). Thus, the interface intersects transversally the boundary of the Lipschitz domain and divides the domain into two Lipschitz sub-domains. First, we use a mixed variational approach to prove the well-posedness of linear problems related to the anisotropic Stokes system. Then we show the existence of a weak solution to the Dirichlet and Dirichlet-transmission problems for the nonlinear anisotropic Navier-Stokes system. This is done by implementing the Leray-Schauder fixed point theorem and using various results and estimates from the linear case, as well as the Leray-Hopf and some other norm inequalities. Explicit conditions for uniqueness of solutions to the nonlinear problems are also provided.

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Section: Introductionmentioning

confidence: 99%

Non-homogeneous Dirichlet-transmission problems for the anisotropic Stokes and Navier-Stokes systems in Lipschitz domains with transversal interfaces

2022

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“…In our case the solvability analysis follows by a combination of Banach fixed-point theory using the velocity as fixed-point variable, and classical Babuška-Brezzi theory for saddle-point problems (by grouping together the velocity and vorticity unknowns). For this we have drawn inspiration from the analysis of Navier-Stokes-Darcy from the recent work [31]. The second aim of this paper is to construct a family of conforming discretisations.…”

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confidence: 99%

Augmented finite element formulation for the Navier--Stokes equations with vorticity and variable viscosity

Anaya¹,

Caraballo²,

Ruiz–Baier³

et al. 2021

Preprint

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We propose and analyse an augmented mixed finite element method for the Navier-Stokes equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and no-slip boundary conditions. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition, and we use a fixed point strategies to show the existence and uniqueness of continuous and discrete solutions under the assumption of sufficiently small data. The method is constructed using any compatible finite element pair for velocity and pressure as dictated by Stokes inf-sup stability, while for vorticity any generic discrete space (of arbitrary order) can be used. We establish optimal a priori error estimates. Finally, we provide a set of numerical tests in 2D and 3D illustrating the behaviour of the scheme as well as verifying the theoretical convergence rates.

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“…We will employ the so-called augmented formulations (also known as Galerkin least-squares methods), which can be regarded as a stabilisation technique where some terms are added to the variational formulation. Augmented finite elements have been considered in several works with applications in fluid mechanics (see, e.g., [8,9,13,20,17,18,34,43] and the references therein). These methods enjoy appealing advantages as those described in length in, e.g., [14,16], and reformulations of the set of equations following this approach are also of great importance in the design of block preconditioners (see [10,31] for an application in Oseen and Navier-Stokes equations in primal form, [30] for stress-velocity-pressure formulations for non-Newtonian flows, or [19,29] for stress-displacement-pressure mixed formulations for hyperelasticity).…”

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confidence: 99%

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

Anaya,

Caraballo,

Gomez-Vargas

et al. 2021

Preprint

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We propose and analyse an augmented mixed finite element method for the Oseen equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and homogeneous Dirichlet boundary condition for the velocity. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition, and we show that it satisfies the hypotheses of the Babuška-Brezzi theory. Repeating the arguments of the continuous analysis, the stability and solvability of the discrete problem are established. The method is suited for any Stokes inf-sup stable finite element pair for velocity and pressure, while for vorticity any generic discrete space (of arbitrary order) can be used. A priori and a posteriori error estimates are derived using two specific families of discrete subspaces. Finally, we provide a set of numerical tests illustrating the behaviour of the scheme, verifying the theoretical convergence rates, and showing the performance of the adaptive algorithm guided by residual a posteriori error estimation.

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A five-field augmented fully-mixed finite element method for the Navier–Stokes/Darcy coupled problem

Cited by 6 publications

References 26 publications

Non-homogeneous Dirichlet-transmission problems for the anisotropic Stokes and Navier-Stokes systems in Lipschitz domains with transversal interfaces

Non-homogeneous Dirichlet-transmission problems for the anisotropic Stokes and Navier-Stokes systems in Lipschitz domains with transversal interfaces

Augmented finite element formulation for the Navier--Stokes equations with vorticity and variable viscosity

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

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