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

Camaño, Jessika; Oyarzúa, Ricardo; Tierra, Giordano

doi:10.1090/mcom/3124

Cited by 30 publications

(29 citation statements)

References 35 publications

(35 reference statements)

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“…Next, proceeding similarly as in (see also ), that is defining now the tensor

σ : = μ (| e (u) |) e (u) - (u \otimes u) - p I in Ω

using the incompressibility and the foregoing equation to eliminate the pressure, introducing the auxiliary unknowns

t : = e (u) and ρ : = \nabla u - e (u),

which denote the strain and the vorticity, respectively, and observing from (2.5) that

bold-italicσ

is now required to be symmetric, which improves the approach from , we arrive at the following system of equations with unknowns

t, u, σ

, and

bold-italicρ

\begin{matrix} \nabla u & = & t + ρ & in Ω, \\ μ (| t |) t - {(u \otimes u)}^{d} & = & {bold-italicσ}^{d} & in Ω, \\ - d i v σ & = & bold-italicf & in Ω, \\ <...> \end{matrix}

…”

Section: The Model Problemmentioning

confidence: 99%

“…We remark in advance that, differently from , Lemma 4.3, Theorem 4.4], where a Strang‐type lemma was used for its proof, we now follow a more straightforward approach, which is based on a suitable decomposition of the error and the Galerkin orthogonality condition. This alternative argumentation can be utilized for the error analysis of other nonlinear problems as well (see, e.g., ).…”

Section: The Galerkin Schemementioning

confidence: 99%

See 1 more Smart Citation

An augmented stress‐based mixed finite element method for the steady state Navier‐Stokes equations with nonlinear viscosity

Camaño

Gatica

Oyarzúa

et al. 2017

Numerical Methods Partial

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A new stress‐based mixed variational formulation for the stationary Navier‐Stokes equations with constant density and variable viscosity depending on the magnitude of the strain tensor, is proposed and analyzed in this work. Our approach is a natural extension of a technique applied in a recent paper by some of the authors to the same boundary value problem but with a viscosity that depends nonlinearly on the gradient of velocity instead of the strain tensor. In this case, and besides remarking that the strain‐dependence for the viscosity yields a more physically relevant model, we notice that to handle this nonlinearity we now need to incorporate not only the strain itself but also the vorticity as auxiliary unknowns. Furthermore, similarly as in that previous work, and aiming to deal with a suitable space for the velocity, the variational formulation is augmented with Galerkin‐type terms arising from the constitutive and equilibrium equations, the relations defining the two additional unknowns, and the Dirichlet boundary condition. In this way, and as the resulting augmented scheme can be rewritten as a fixed‐point operator equation, the classical Schauder and Banach theorems together with monotone operators theory are applied to derive the well‐posedness of the continuous and associated discrete schemes. In particular, we show that arbitrary finite element subspaces can be utilized for the latter, and then we derive optimal a priori error estimates along with the corresponding rates of convergence. Next, a reliable and efficient residual‐based a posteriori error estimator on arbitrary polygonal and polyhedral regions is proposed. The main tools used include Raviart‐Thomas and Clément interpolation operators, inverse and discrete inequalities, and the localization technique based on triangle‐bubble and edge‐bubble functions. Finally, several numerical essays illustrating the good performance of the method, confirming the reliability and efficiency of the a posteriori error estimator, and showing the desired behavior of the adaptive algorithm, are reported. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1692–1725, 2017

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“…Next, proceeding similarly as in (see also ), that is defining now the tensor

σ : = μ (| e (u) |) e (u) - (u \otimes u) - p I in Ω

using the incompressibility and the foregoing equation to eliminate the pressure, introducing the auxiliary unknowns

t : = e (u) and ρ : = \nabla u - e (u),

which denote the strain and the vorticity, respectively, and observing from (2.5) that

bold-italicσ

is now required to be symmetric, which improves the approach from , we arrive at the following system of equations with unknowns

t, u, σ

, and

bold-italicρ

\begin{matrix} \nabla u & = & t + ρ & in Ω, \\ μ (| t |) t - {(u \otimes u)}^{d} & = & {bold-italicσ}^{d} & in Ω, \\ - d i v σ & = & bold-italicf & in Ω, \\ <...> \end{matrix}

…”

Section: The Model Problemmentioning

confidence: 99%

Section: The Galerkin Schemementioning

confidence: 99%

An augmented stress‐based mixed finite element method for the steady state Navier‐Stokes equations with nonlinear viscosity

Camaño

Gatica

Oyarzúa

et al. 2017

Numerical Methods Partial

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“…It is easy to see that they both satisfy (2.3) with (μ 1 , μ 2 ) = (2, 3) and (μ 1 , μ 2 ) = (α 0 , α 0 + α 1 ), respectively. Next, following [13] and [16], we observe that the first equation in (2.1) can be rewritten as the equilibrium equation…”

Section: The Navier-stokes Equations With Variable Viscositymentioning

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.

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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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“…More precisely, the approach in [11] employs a technique previously applied to the Navier-Stokes equations with constant viscosity (see [12] and [32]), which is based on the introduction of a modified pseudostress tensor involving the diffusive and convective terms, and the pressure. The latter is then eliminated thanks to the incompressibility condition, and the nonlinear viscosity is handled by incorporating the gradient of velocity as an auxiliary tensor unknown.…”

Section: Introductionmentioning

confidence: 99%

A posteriori error analysis of an augmented mixed method for the Navier–Stokes equations with nonlinear viscosity

Gatica

Ruiz–Baier

Tierra

2016

Computers & Mathematics with Applications

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In this work we develop the a posteriori error analysis of an augmented mixed finite element method for the 2D and 3D versions of the Navier-Stokes equations when the viscosity depends nonlinearly on the module of the velocity gradient. Two different reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary (convex or non-convex) polygonal and polyhedral regions are derived. Our analysis of reliability of the proposed estimators draws mainly upon the global inf-sup condition satisfied by a suitable linearization of the continuous formulation, an application of Helmholtz decomposition, and the local approximation properties of the RaviartThomas and Clément interpolation operators. In addition, differently from previous approaches for augmented mixed formulations, the boundedness of the Clément operator plays now an interesting role in the reliability estimate. On the other hand, inverse and discrete inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show their efficiency. Finally, several numerical results are provided to illustrate the good performance of the augmented mixed method, to confirm the aforementioned properties of the a posteriori error estimators, and to show the behaviour of the associated adaptive algorithm.

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

Cited by 30 publications

References 35 publications

An augmented stress‐based mixed finite element method for the steady state Navier‐Stokes equations with nonlinear viscosity

An augmented stress‐based mixed finite element method for the steady state Navier‐Stokes equations with nonlinear viscosity

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

A posteriori error analysis of an augmented mixed method for the Navier–Stokes equations with nonlinear viscosity

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