On stabilized mixed methods for generalized Stokes problem based on the velocity–pseudostress formulation: A priori error estimates

Barrios, Tomás P.; Bustinza, Rommel; García, Galina C.; Hernández, Erwin

doi:10.1016/j.cma.2012.05.006

Cited by 19 publications

(16 citation statements)

References 25 publications

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“…In , the pseudostress and the trace‐free velocity gradient are introduced as auxiliary unknowns, and a pseudostress–velocity formulation is considered, for which existence, uniqueness, and error estimates are derived. More recently, dual‐mixed methods based on the velocity–pseudostress and pseudostress have been introduced in and , respectively, for the generalized Stokes problem. In the former, the approach from (see also ) is adapted to propose an augmented mixed method in terms of velocity and pseudostress, for which optimal error estimates are proved.…”

Section: Introductionmentioning

confidence: 99%

“…In the former, the approach from (see also ) is adapted to propose an augmented mixed method in terms of velocity and pseudostress, for which optimal error estimates are proved. On the other hand, in , a formulation based only on the pseudostress is proposed for the Brinkman problem, thus simplifying and improving the analysis from . The results in include a priori and a posteriori error analyses of the resulting Galerkin scheme.…”

Section: Introductionmentioning

confidence: 99%

“…The model problem. The Brinkman (or generalized Stokes) problem , formulated in terms of the velocity u , vorticity ω , and pressure p of an incompressible viscous fluid (see also ), reads as follows: given a force density f and a vector field a , we seek a vector field u , a scalar field ω , and a scalar field p such that

({, \begin{matrix} σ u + ν curl ω + \nabla p & = & f & in Ω, \\ ω - rot u & = & 0 & in Ω, \\ div u & = & 0 & in Ω, \\ u \cdot t & = & a \cdot t & on Σ, \\ p & = & 0 & on Σ, \\ u \cdot n & = & 0 & on Γ, \\ ω & = & 0 & on Γ, \end{matrix})

where u · t and u · n stand for the normal and tangential components of the velocity, respectively. In the model, ν > 0 is the kinematic viscosity of the fluid, and σ > 0 is a parameter representing the ratio between the fluid density and the permeability of the porous medium.…”

Section: Introductionmentioning

confidence: 99%

“…However, up to our knowledge, the Brinkman problem has been considered using mixed vorticity-velocitypressure formulations only very recently [19], in where a dual-mixed formulation has been analyzed at the continuous and discrete levels using the Babuška-Brezzi theory and optimal error estimates are provided.The so-called augmented mixed finite elements (also known as Galerkin least-squares methods [12,20,21]) can be regarded as a stabilization technique where some terms are added to the variational formulation so that, either the resulting augmented variational formulations are defined by strongly coercive bilinear forms (see, e.g., [22]), or they enable to bypass the kernel property, which is very difficult to obtain in practice, or they allow the fulfillment of the inf-sup condition at the continuous and discrete levels in mixed formulations ([23]). This approach has been considered in, for example, [5,[24][25][26][27][28][29] for Stokes, generalized Stokes (in velocity-pseudostress formulation), coupling of quasi-Newtonian fluids and porous media, and Navier-Stokes equations, and in [30] for an augmented mixed formulation applied to elliptic problems with mixed boundary conditions.Here, we propose a new class of stabilized finite element approximations of the Brinkman equations, written in terms of the velocity, vorticity, and pressure fields. One of the main goals of the present approach is to build different families of finite elements to approximate the model problem with the liberty of choosing any combination of the finite element subspaces of the continuous spaces and extend recent results given in [24], where a new stabilized finite element approximation for the Stokes equations was analyzed using an extension of the Babuška-Brezzi theory (cf.…”

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An augmented velocity–vorticity–pressure formulation for the Brinkman equations

Anaya

Gatica

Mora

et al. 2015

Numerical Methods in Fluids

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SUMMARYThis paper deals with the analysis of a new augmented mixed finite element method in terms of vorticity, velocity and pressure, for the Brinkman problem with non-standard boundary conditions. The approach is based on the introduction of Galerkin least-squares terms arising from the constitutive equation relating the aforementioned unknowns, and from the incompressibility condition. We show that the resulting augmented bilinear form is continuous and elliptic which, thanks to the Lax-Milgram Theorem, and besides proving the well-posedness of the continuous formulation, ensures the solvability and stability of the Galerkin scheme with any finite element subspace of the continuous space. In particular, Raviart-Thomas elements of any order k ≥ 0 for the velocity field, and piecewise continuous polynomials of degree k + 1 for both the vorticity and the pressure, can be utilized. A priori error estimates and the corresponding rates of convergence are also given here. Next, we derive two reliable and efficient residual-based a posteriori error estimators for this problem. The ellipticity of the bilinear form together with the local approximation properties of the Clément interpolation operator are the main tools for showing the reliability. In turn, inverse inequalities and the localization technique based on triangle-bubble and edge-bubble functions are utilized to show the efficiency. Finally, several numerical results illustrating the good performance of the method, confirming the properties of the estimators, and showing the behaviour of the associated adaptive algorithms, are reported. Copyright c 2014 John Wiley & Sons, Ltd.Received . . . KEY WORDS: Generalized Stokes equations; Brinkman problem; vorticity-based mixed formulation; augmented scheme; finite element method; a priori error estimates; a posteriori error analysis.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

({, \begin{matrix} σ u + ν curl ω + \nabla p & = & f & in Ω, \\ ω - rot u & = & 0 & in Ω, \\ div u & = & 0 & in Ω, \\ u \cdot t & = & a \cdot t & on Σ, \\ p & = & 0 & on Σ, \\ u \cdot n & = & 0 & on Γ, \\ ω & = & 0 & on Γ, \end{matrix})

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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An augmented velocity–vorticity–pressure formulation for the Brinkman equations

Anaya

Gatica

Mora

et al. 2015

Numerical Methods in Fluids

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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%

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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A stabilized mixed method applied to Stokes system with nonhomogeneous source terms: The stationary case Dedicated to Prof. R. Rodríguez, on the occasion of his 65th birthday

Barrios

Behrens

Bustinza

2019

Numerical Methods in Fluids

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This article is concerned with the Stokes system with nonhomogeneous source terms and nonhomogeneous Dirichlet boundary condition. First, we reformulate the problem in its dual mixed form, and then, we study its corresponding well-posedness. Next, in order to circumvent the well-known Babuška-Brezzi condition, we analyze a stabilized formulation of the resulting approach. Additionally, we endow the scheme with an a posteriori error estimator that is reliable and efficient. Finally, we provide numerical experiments that illustrate the performance of the corresponding adaptive algorithm and support its use in practice. KEYWORDSa posteriori error estimates, augmented mixed formulation, Ritz projection of the error INTRODUCTIONIn the work of Cai et al, 1 a dual mixed finite element method for the incompressible fluid flow was introduced and analyzed. The approach there follows the ideas developed in the other work of Cai et al, 2 ie, the incompressible fluid flow is reformulated using the new variable so-called pseudostress, which is in relation with the pressure and gradient of the velocity. The main advantage of this new variable is the accurate approximation to physical quantities such as the stress and vorticity, allowing to use the pair of conforming Raviart-Thomas with discontinuous polynomial as the finite element space. Furthermore, in order to obtain more flexibility in the finite element spaces, the stabilization of this approach has been studied in the work of Figueroa et al. 3 In addition, its corresponding extension to quasi Newtonian flows and Brinkman model were developed in the works of Gatica et al 4 and Barrios et al, 5 respectively. On the other hand, concerning linear elasticity problem, in the work of Barrios et al, 6 the authors present an alternative a posteriori error estimator to the previous one developed in a former work. 7 This approach is based on the Ritz projection Int J Numer Meth Fluids. 2020;92:509-527. wileyonlinelibrary.com/journal/fld

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On stabilized mixed methods for generalized Stokes problem based on the velocity–pseudostress formulation: A priori error estimates

Cited by 19 publications

References 25 publications

An augmented velocity–vorticity–pressure formulation for the Brinkman equations

An augmented velocity–vorticity–pressure formulation for the Brinkman equations

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

A stabilized mixed method applied to Stokes system with nonhomogeneous source terms: The stationary case Dedicated to Prof. R. Rodríguez, on the occasion of his 65th birthday

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