Augmented mixed finite element methods for a vorticity‐based velocity–pressure–stress formulation of the Stokes problem in 2D

Gatica, Gabriel N.; Gatica, Luis F.; Márquez, Antonio

doi:10.1002/fld.2362

Cited by 13 publications

(15 citation statements)

References 14 publications

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“…In this way, we proceed as in the proof of Lemma 3.3, and utilize the continuity of C and the fact that φ 1 ∈ K, to obtain 20) which together to assumption (3.17) implies that J is a contraction mapping. Hence, applying the Banach's fixed point Theorem we obtain that there exists a unique φ ∈ K such that J (φ) = φ, or equivalently, there exists a unique φ ∈ X solution to (3.3), which satisfies (3.18).…”

Section: The Main Resultsmentioning

confidence: 99%

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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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Section: The Main Resultsmentioning

confidence: 99%

“…[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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“…Several numerical methods exploit these properties, as for instance, different formulations based on least-squares, stabilization techniques, mixed finite elements, spectral discretizations, and hybridizable discontinuous Galerkin methods (see for instance [3,4,8,12,14,18,19,[21][22][23]26,[34][35][36], and the references therein). For the generalized Stokes problem written in velocity-vorticity-pressure variables, we mention [6] where an augmented mixed formulation based on RT k − P k+1 − P k+1 (with continuous pressure approximation) finite elements has been developed and analyzed.…”

Section: Introductionmentioning

confidence: 99%

A priori and a posteriori error analysis of a mixed scheme for the Brinkman problem

et al. 2015

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This paper deals with the analysis of new mixed finite element methods for the Brinkman equations formulated in terms of velocity, vorticity and pressure. Employing the Babuška-Brezzi theory, it is proved that the resulting continuous and discrete variational formulations are well-posed. In particular, we show that RaviartThomas elements of order k ≥ 0 for the approximation of the velocity field, piecewise continuous polynomials of degree k + 1 for the vorticity, and piecewise polynomials of degree k for the pressure, yield unique solvability of the discrete problem. On the other hand, we also show that families of finite elements based on Brezzi-DouglasMarini elements of order k +1 for the approximation of velocity, piecewise continuous polynomials of degree k +2 for the vorticity, and piecewise polynomials of degree k for the pressure ensure the well-posedness of the associated Galerkin scheme. We note that these families provide exactly divergence-free approximations of the velocity field. We establish a priori error estimates in the natural norms with constants independent B Ricardo Ruiz-Baier

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“…This approach has been considered in e.g. [5,23,24,29,34,38] for Stokes, generalized Stokes, and Navier-Stokes equations 0045-7825/$ -see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cma.2013.08.011 and in [6] for an augmented mixed formulation applied to elliptic problems with mixed boundary conditions, whereas other related methods for the vorticity-velocity-pressure formulation based on least-squares, spectral discretization, hybridizable discontinuous Galerkin, can be found in [4,8,10,[16][17][18]20,39], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…Among the available results in the context of mixed finite elements for vorticity-based formulations, we mention the P 0 À P 1 À P 0 formulation introduced in [3], and the augmented formulation in [29], written also in terms of stresses ( [14,15]). A somewhat different approach has been presented in [22], where the problem is written as a system of first order equations and the resulting variables are discretized in terms of P 1 À RT 0 À P 0 elements.…”

Section: Introductionmentioning

confidence: 99%

An augmented mixed finite element method for the vorticity–velocity–pressure formulation of the Stokes equations

Anaya

Mora

Ruiz–Baier

2013

Computer Methods in Applied Mechanics and Engineering

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a b s t r a c tThis paper deals with the numerical approximation of the stationary two-dimensional Stokes equations, formulated in terms of vorticity, velocity and pressure, with non-standard boundary conditions. Here, by introducing a Galerkin least-squares term, we end up with a stabilized variational formulation that can be recast as a twofold saddle point problem. We propose two families of mixed finite elements to solve the discrete problem, in the first family, the unknowns are approximated by piecewise continuous and quadratic elements, Brezzi-Douglas-Marini, and piecewise constant finite elements, respectively, while in the second family, the unknowns are approximated by piecewise linear and continuous, Raviart-Thomas, and piecewise constant finite elements, respectively. The wellposedness of the resulting continuous and discrete variational problems are studied employing an extension of the Babuška-Brezzi theory. We establish a priori error estimates in the natural norms, and we finally report some numerical experiments illustrating the behavior of the proposed schemes and confirming our theoretical findings on structured and unstructured meshes. Additional examples of cases not covered by our theory are also presented.

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Augmented mixed finite element methods for a vorticity‐based velocity–pressure–stress formulation of the Stokes problem in 2D

Cited by 13 publications

References 14 publications

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

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

A priori and a posteriori error analysis of a mixed scheme for the Brinkman problem

An augmented mixed finite element method for the vorticity–velocity–pressure formulation of the Stokes equations

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