An augmented mixed finite element method with Lagrange multipliers: A priori and a posteriori error analyses

Barrios, Tomás P.; Gatica, Gabriel N.

doi:10.1016/j.cam.2006.01.017

Cited by 18 publications

(20 citation statements)

References 16 publications

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“…A recent alternative to the least-squares mixed finite element method, also with the goal to avoid the Babuška-Brezzi condition, was given in [1]. In that paper it was proposed to add to the mixed weak formulation (11) the equalities…”

Section: A Non-standard Mixed Finite Element Methodsmentioning

confidence: 99%

“…Nonetheless, in [1] it was shown that C is both continuous and coercive, hence the discrete problem is well-posed.…”

Section: A Non-standard Mixed Finite Element Methodsmentioning

confidence: 99%

“…Using an elliptic model problem, we introduce the mixed finite element method and its non-standard variant from [1]. We also recall the least-squares mixed finite element method, since it is our objective to compare it with the non-standard method.…”

Section: Preliminariesmentioning

confidence: 99%

“…In a recent paper [1], the authors proposed an alternative way to circumvent the Babuška-Brezzi conditions that complicate the mixed finite element method. By adding two terms to the mixed formulation, they arrived at a new, non-standard mixed finite element method.…”

Section: Introductionmentioning

confidence: 99%

“…By adding two terms to the mixed formulation, they arrived at a new, non-standard mixed finite element method. The current paper aims to analyze the approximation porperties of this method in a way different from that in [1]. By comparing the new non-standard approximations to those obtained from the least-squares mixed finite element method directly, the conclusion is the same as for this least-squares mixed method: when classical examples of finite element spaces are used and the model problem at hand is well-behaved, the non-standard method is a higher order perturbation of the standard and the mixed method.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of a non-standard mixed finite element method with applications to superconvergence

Brandts

2009

Appl Math

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Section: A Non-standard Mixed Finite Element Methodsmentioning

confidence: 99%

“…Nonetheless, in [1] it was shown that C is both continuous and coercive, hence the discrete problem is well-posed.…”

Section: A Non-standard Mixed Finite Element Methodsmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analysis of a non-standard mixed finite element method with applications to superconvergence

Brandts

2009

Appl Math

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

Gatica

Márquez

2011

Numerical Methods in Fluids

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SUMMARYIn this paper, we consider an augmented velocity-pressure-stress formulation of the 2D Stokes problem, in which the stress is defined in terms of the vorticity and the pressure, and then we introduce and analyze stable mixed finite element methods to solve the associated Galerkin scheme. In this way, we further extend similar procedures applied recently to linear elasticity and to other mixed formulations for incompressible fluid flows. Indeed, our approach is based on the introduction of the Galerkin leastsquares-type terms arising from the corresponding constitutive and equilibrium equations, and from the Dirichlet boundary condition for the velocity, all of them multiplied by stabilization parameters. Then, we show that these parameters can be suitably chosen so that the resulting operator equation induces a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. In particular, we can use continuous piecewise linear velocities, piecewise constant pressures, and rotated Raviart-Thomas elements for the stresses. Next, we derive reliable and efficient residual-based a posteriori error estimators for the augmented mixed finite element schemes. In addition, several numerical experiments illustrating the performance of the augmented mixed finite element methods, confirming the properties of the a posteriori estimators, and showing the behavior of the associated adaptive algorithms are reported. The present work should be considered as a first step aiming finally to derive augmented mixed finite element methods for vorticity-based formulations of the 3D Stokes problem.

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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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An augmented mixed finite element method with Lagrange multipliers: A priori and a posteriori error analyses

Cited by 18 publications

References 16 publications

Analysis of a non-standard mixed finite element method with applications to superconvergence

Analysis of a non-standard mixed finite element method with applications to superconvergence

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

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

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