An augmented mixed-primal finite element method for a coupled flow-transport problem

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

doi:10.1051/m2an/2015015

Cited by 38 publications

(44 citation statements)

References 27 publications

(43 reference statements)

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“…We will go back to this fact later on in section 3. In addition, it is easy to see that the forthcoming analysis also applies to the slightly more general case of a viscosity function acting on Ω × R + , that is μ : Ω × R + −→ R. Some examples of nonlinear μ are the following: where α 0 , α 1 > 0 and β ∈ [1,2]. The first example is basically academic but the second one corresponds to a particular case of the well-known Carreau law in fluid mechanics.…”

Section: The Navier-stokes Equations With Variable Viscositymentioning

confidence: 99%

“…However, we prefer to keep the above analysis as it is since, being much more general, it provides a quite useful logical sequence for studying similar and related problems. Indeed, in most of the solvability analyses of more involved fixed point equations, a second condition on the data, different from the one ensuring that the corresponding operator maps a given closed and convex domain into itself, is required for the uniqueness of solutions (see, e.g., [2] for a recent work in this direction concerning a coupled flow-transport problem). The fact that the same condition on the data guarantees both existence and uniqueness of the solution might very well be a particular feature of the present problem and its associated fixed point operator T.…”

Section: ω) This Proves That T(w ρ ) Is Compact and Finishes The Proofmentioning

confidence: 99%

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An Augmented Mixed Finite Element Method for the Navier--Stokes Equations with Variable Viscosity

Camaño¹,

Gatica²,

Oyarzúa³

et al. 2016

SIAM J. Numer. Anal.

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Abstract.A new mixed variational formulation for the Navier-Stokes equations with constant density and variable viscosity depending nonlinearly on the gradient of velocity, is proposed and analyzed here. Our approach employs a technique previously applied to the stationary Boussinesq problem and to the Navier-Stokes equations with constant viscosity, which consists firstly of the introduction of a modified pseudostress tensor involving the diffusive and convective terms, and the pressure. Next, by using an equivalent statement suggested by the incompressibility condition, the pressure is eliminated, and in order to handle the nonlinear viscosity, the gradient of velocity is incorporated as an auxiliary unknown. Furthermore, since the convective term forces the velocity to live in a smaller space than usual, we overcome this difficulty by augmenting the variational formulation with suitable Galerkin-type terms arising from the constitutive and equilibrium equations, the aforementioned relation defining the additional unknown, and the Dirichlet boundary condition. The resulting augmented scheme is then written equivalently as a fixed point equation, and hence the well-known Schauder and Banach theorems, combined with classical results on bijective monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. No discrete inf-sup conditions are required for the well-posedness of the Galerkin scheme, and hence arbitrary finite element subspaces of the respective continuous spaces can be utilized. In particular, given an integer k ≥ 0, piecewise polynomials of degree ≤ k for the gradient of velocity, Raviart-Thomas spaces of order k for the pseudostress, and continuous piecewise polynomials of degree ≤ k + 1 for the velocity, constitute feasible choices. Finally, optimal a priori error estimates are derived, and several numerical results illustrating the good performance of the augmented mixed finite element method and confirming the theoretical rates of convergence are reported.

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Section: The Navier-stokes Equations With Variable Viscositymentioning

confidence: 99%

Section: ω) This Proves That T(w ρ ) Is Compact and Finishes The Proofmentioning

confidence: 99%

An Augmented Mixed Finite Element Method for the Navier--Stokes Equations with Variable Viscosity

Camaño¹,

Gatica²,

Oyarzúa³

et al. 2016

SIAM J. Numer. Anal.

Self Cite

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“…This result will be attained by using a fixed-point strategy considering a similar approach as in [12, Section 3.2] (see also [3,16]). …”

Section: Analysis Of the Continuous Navier-stokes/darcy Problemmentioning

confidence: 99%

A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

Discacciati

Oyarzúa

2016

Numer. Math.

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Citation: DISCACCIATI, M. and OYARZUA, R., 2016. A conforming mixed finite element method for the Navier Stokes/Darcy coupled problem. Numerische Mathematik, 135(2), pp. 571 606. Abstract In this paper we develop the a priori analysis of a mixed finite element method for the coupling of fluid flow with porous media flow. Flows are governed by the Navier-Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. We consider the standard mixed formulation in the Navier-Stokes domain and the dual-mixed one in the Darcy region, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The finite element subspaces defining the discrete formulation employ Bernardi-Raugel and RaviartThomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for the Lagrange multiplier. We show stability, convergence, and a priori error estimates for the associated Galerkin scheme. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are reported.

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“…[5], [14], [20], [22], and the references therein). This study has been extended to solve important problems in engineering, such as the Stokes-Darcy coupled problem and transport problems (see, for instance, [1], [21], [24], [25]). …”

Section: Introductionmentioning

confidence: 99%

Analysis of an augmented mixed-FEM for the Navier-Stokes problem

Camaño¹,

Oyarzúa²,

Tierra

2016

Math. Comp.

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In this paper we propose and analyze a new augmented mixed finite element method for the Navier-Stokes problem. Our approach is based on the introduction of a "nonlinearpseudostress" tensor linking the pseudostress tensor with the convective term, which leads to a mixed formulation with the nonlinear-pseudostress tensor and the velocity as the main unknowns of the system. Further variables of interest, such as the fluid pressure, the fluid vorticity and the fluid velocity gradient, can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. The resulting mixed formulation is augmented by introducing Galerkin least-squares type terms arising from the constitutive and equilibrium equations of the Navier-Stokes equations and from the Dirichlet boundary condition, which are multiplied by stabilization parameters that are chosen in such a way that the resulting continuous formulation becomes well-posed. Then, the classical Banach's fixed point Theorem and Lax-Milgram's Lemma are applied to prove well-posedness of the continuous problem. Similarly, we establish well-posedness and the corresponding Cea's estimate of the associated Galerkin scheme considering any conforming finite element subspace for each unknown. In particular, the associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree k for the nonlinear-pseudostress tensor, and continuous piecewise polynomial elements of degree k + 1 for the velocity, which leads to an optimal convergent scheme. In addition, we provide two iterative methods to solve the corresponding nonlinear system of equations and analyze their convergence. Finally, several numerical results illustrating the good performance of the method are provided.

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An augmented mixed-primal finite element method for a coupled flow-transport problem

Cited by 38 publications

References 27 publications

An Augmented Mixed Finite Element Method for the Navier--Stokes Equations with Variable Viscosity

An Augmented Mixed Finite Element Method for the Navier--Stokes Equations with Variable Viscosity

A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

Analysis of an augmented mixed-FEM for the Navier-Stokes problem

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