A residual based<i>A POSTERIORI</i>error estimator for an augmented mixed finite element method in linear elasticity

We develop the a posteriori error analysis for a mixed finite element method applied to the coupling of Brinkman and Darcy equations in 3D, modelling the interaction of viscous and non-viscous flow effects across a given interface. The system is formulated in terms of velocity and pressure within the Darcy subdomain, together with vorticity, velocity and pressure of the fluid in the Brinkman region, and a Lagrange multiplier enforcing pressure continuity across the interface. The solvability of a fullymixed formulation along with a priori error bounds for a finite element method have been recently established in Álvarez et al. ( Comput Methods Appl Mech Eng 307:68-95, 2016). Here we derive a residual-based a posteriori error estimator for such a scheme, and prove its reliability exploiting a global inf-sup condition in combination with suitable Helmholtz decompositions, and interpolation properties of Clément and Raviart-Thomas operators. The estimator is also shown to be efficient, following a localisation strategy and appropriate inverse inequalities. We present numerical tests to confirm the features of the estimator and to illustrate the performance of the method in academic and application-oriented problems.

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“…The proof of (3.30) corresponds to a slight adaptation of the proof of [7,Lemma 4.6], which makes use of (3.29).…”

Section: Lemma 16 Let ζ H ∈ L 2 ( ) Be An Element-wise Polynomial Of mentioning

confidence: 99%

A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem

Alvarez

Gatica

Ruiz–Baier

2017

Calcolo

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“…For the proof of (3.50) we refer to [7,Lemma 4.3], whereas (3.51) is a slight modification of the proof of [7,Lemma 4.4]. We omit further details.…”

Section: Lemma 316mentioning

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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“…Note that no extra regularity assumptions on the data, but only f ∈ [L 2 (Ω s )] 2 and g ∈ H −1/2 (Γ ), are needed here. Theorem 3 Let ((σ , p), (ϕ, γ )) ∈ H × Q be the unique solution of (8) and let u ∈ [L 2 (Ω s )] 2 be defined according to (7).…”

Section: The Mixed Variational Formulationmentioning

confidence: 99%

“…In addition, let u and u h be defined according to (7) and (37), respectively, and suppose that the Robin datum g belongs to L 2 (Γ ). Also, assume that there exists c ≥ 1 such thath e ≤ c h e for each e ∈ E h (Σ).…”

Section: Theoremmentioning

confidence: 99%

A residual-based a posteriori error estimator for a two-dimensional fluid–solid interaction problem

2009

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In this paper, we develop an a posteriori error analysis of a mixed finite element method for a fluid-solid interaction problem posed in the plane. The media are governed by the acoustic and elastodynamic equations in time-harmonic regime, respectively, and the transmission conditions are given by the equilibrium of forces and the equality of the normal displacements of the solid and the fluid. The coupling of primal and dual-mixed finite element methods is applied to compute both the pressure of the scattered wave in the linearized fluid and the elastic vibrations that take place in the elastic body. The finite element subspaces consider continuous piecewise linear elements for the pressure and a Lagrange multiplier defined on the interface, and PEERS for the stress and rotation in the solid domain. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. Suitable auxiliary problems, the continuous inf-sup conditions satisfied by the bilinear forms involved, a discrete Helmholtz decomposition, and the local approximation properties 123 64 G. N. Gatica et al. of the Clément interpolant and Raviart-Thomas operator are the main tools for proving the reliability of the estimator. Then, Helmholtz decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are employed to show the efficiency. Finally, some numerical results confirming the reliability and efficiency of the estimator are reported.

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A residual basedA POSTERIORIerror estimator for an augmented mixed finite element method in linear elasticity

Cited by 41 publications

References 12 publications

A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem

A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem

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

A residual-based a posteriori error estimator for a two-dimensional fluid–solid interaction problem

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