Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow

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

doi:10.1007/s00211-013-0577-x

Cited by 54 publications

(44 citation statements)

References 30 publications

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“…It is similar to the proof of [28,Lemma 20]. Given T ∈ T h we denote γ T := r(u h , p h ; f ) in T .…”

Section: )mentioning

confidence: 88%

“…Similarly as in the proof of [28,Lemma 23], it suffices to reconsider the local estimate (5.44), and observe that, as a consequence of the Cauchy-Schwarz inequality, the last term of it is bounded by d ds (λ − λ s ) 0,e γ 0,e . The rest follows exactly as in the last part of the proof of Lemma 5.15.…”

Section: )mentioning

confidence: 94%

“…To this end, we proceed as in [17] and [28], and apply the localization technique based on triangle-bubble and edge-bubble functions, together with extension operators, discrete trace and inverse inequalities. Therefore, we now introduce further notations and preliminary results.…”

Section: Efficiency Of the A Posteriori Error Estimatormentioning

confidence: 99%

“…We proceed as in the proof of [28,Lemma 21]. In fact, given e ∈ E h (Γ D ) we denote γ e := r(u h , p h ; f )·s on e. Since p = 0 on Γ D (cf.…”

Section: )mentioning

confidence: 99%

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A mixed finite element method for Darcy’s equations with pressure dependent porosity

2015

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In this work we develop the a priori and a posteriori error analyses of a mixed finite element method for Darcy's equations with porosity depending exponentially on the pressure. A simple change of variable for this unknown allows to transform the original nonlinear problem into a linear one whose dual-mixed variational formulation falls into the frameworks of the generalized linear saddle point problems and the fixed point equations satisfied by an affine mapping. According to the latter, we are able to show the well-posedness of both the continuous and discrete schemes, as well as the associated Cea estimate, by simply applying a suitable combination of the classical Babuška-Brezzi theory and the Banach fixed point Theorem. In particular, given any integer k ≥ 0, the stability of the Galerkin scheme is guaranteed by employing Raviart-Thomas elements of order k for the velocity, piecewise polynomials of degree k for the pressure, and continuous piecewise polynomials of degree k+1 for an additional Lagrange multiplier given by the trace of the pressure on the Neumann boundary. Note that the two ways of writing the continuous formulation suggest accordingly two different methods for solving the discrete schemes. Next, we derive a reliable and efficient residualbased a posteriori error estimator for this problem. The global inf-sup condition satisfied by the continuous formulation, Helmholtz decompositions, and the local approximation properties of the Raviart-Thomas and Clément interpolation operators are the main tools for proving the reliability. In turn, inverse and discrete 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 both methods, confirming the aforementioned properties of the estimator, and showing the behaviour of the associated adaptive algorithm, are reported.

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“…It is similar to the proof of [28,Lemma 20]. Given T ∈ T h we denote γ T := r(u h , p h ; f ) in T .…”

Section: )mentioning

confidence: 88%

Section: )mentioning

confidence: 94%

Section: Efficiency Of the A Posteriori Error Estimatormentioning

confidence: 99%

“…We proceed as in the proof of [28,Lemma 21]. In fact, given e ∈ E h (Γ D ) we denote γ e := r(u h , p h ; f )·s on e. Since p = 0 on Γ D (cf.…”

Section: )mentioning

confidence: 99%

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A mixed finite element method for Darcy’s equations with pressure dependent porosity

2015

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“…(3.4)). For this purpose, we proceed as in [13,24,4], and apply the localization technique based on triangle-bubble and edge-bubble functions, together with extension operators, discrete trace and inverse inequalities. According to the above, we now introduce additional notations and preliminary results.…”

Section: Efficiency Of the A Posteriori Error Estimatorsmentioning

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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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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Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow

Cited by 54 publications

References 30 publications

A mixed finite element method for Darcy’s equations with pressure dependent porosity

A mixed finite element method for Darcy’s equations with pressure dependent porosity

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

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

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