A Posteriori Error Estimation for an Interior Penalty Type Method Employing $H(\mathrm{div})$ Elements for the Stokes Equations

Wang, Junping; Wang, Yanqiu; Ye, Xiu

doi:10.1137/100783996

Cited by 9 publications

(4 citation statements)

References 23 publications

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“…We shall focus our attention on residual type a posteriori error estimators, in which the computable formula for judging the efficiency and reliability of numerical schemes is given as functions of residuals. Along this avenue, many fine results have been developed for finite element methods for the Stokes equations [16][17][18][19][20][21][22][23][24][25][26][27]. However, little can be seen in the existing literature for the finite volume methods for Stokes equations.…”

Section: Introductionmentioning

confidence: 99%

A unified a posteriori error estimator for finite volume methods for the stokes equations

Wang

2013

Math Methods in App Sciences

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In this paper, the authors established a unified framework for deriving and analyzing a posteriori error estimators for finite volume methods for the Stokes equations. The a posteriori error estimators are residual based and are applicable to various finite volume methods for the Stokes equations. In particular, the unified theoretical analysis works well for finite volume schemes arising from using trial functions of conforming, nonconforming, and discontinuous finite element functions, yielding new results that are not seen in the existing literature. Copyright © 2013 John Wiley & Sons, Ltd.

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Section: Introductionmentioning

confidence: 99%

A unified a posteriori error estimator for finite volume methods for the stokes equations

Wang

2013

Math Methods in App Sciences

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“…A posteriori error estimations have been widely studied for both the mixed formulations of the Darcy flow [24][25][26] and the Stokes flow [27][28][29][30][31][32][33][34][35][36]. However, only a few works exist for the coupled Darcy-Stokes problem, see for instance [18,[37][38][39][40].…”

Section: Introductionmentioning

confidence: 99%

A Posteriori Error Analysis for a New Fully Mixed Isotropic Discretization of the Stationary Stokes-Darcy Coupled Problem

Houédanou

Adetola

Allaoui³

2020

Abstract and Applied Analysis

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In this paper, we develop an a posteriori error analysis for the stationary Stokes-Darcy coupled problem approximated by conforming the finite element method on isotropic meshes in ℝ d , d ∈ 2 , 3 . The approach utilizes a new robust stabilized fully mixed discretization. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution plus the stabilization terms. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient.

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“…Suitable also for low order elements is the enrichment of the velocity space by rational divergence-free functions [23,24]. Using dG-approximations for the tangential components of the vector Laplacian, H(div)-conforming divergence-free methods can be constructed [11,27,29,42]. Furthermore, isogeometric analysis allows to define robust mixed methods with pressure-independent velocity errors [10,15].…”

Section: Introductionmentioning

confidence: 99%

Robust Arbitrary Order Mixed Finite Element Methods for the Incompressible Stokes Equations with pressure independent velocity errors

2016

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Standard mixed finite element methods for the incompressible Navier-Stokes equations that relax the divergence constraint are not robust against large irrotational forces in the momentum balance and the velocity error depends on the continuous pressure. This robustness issue can be completely cured by using divergence-free mixed finite elements which deliver pressure-independent velocity error estimates. However, the construction of H 1 -conforming, divergence-free mixed finite element methods is rather difficult. Instead, we present a novel approach for the construction of arbitrary order mixed finite element methods which deliver pressure-independent velocity errors. The approach does not change the trial functions but replaces discretely divergence-free test functions in some operators of the weak formulation by divergence-free ones. This modification is applied to inf-sup stable conforming and nonconforming mixed finite element methods of arbitrary order in two and three dimensions. Optimal estimates for the incompressible Stokes equations are proved for the H 1 and L 2 errors of the velocity and the L 2 error of the pressure. Moreover, both velocity errors are pressure-independent, demonstrating the improved robustness. Several numerical examples illustrate the results.

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A Posteriori Error Estimation for an Interior Penalty Type Method Employing $H(\mathrm{div})$ Elements for the Stokes Equations

Cited by 9 publications

References 23 publications

A unified a posteriori error estimator for finite volume methods for the stokes equations

A unified a posteriori error estimator for finite volume methods for the stokes equations

A Posteriori Error Analysis for a New Fully Mixed Isotropic Discretization of the Stationary Stokes-Darcy Coupled Problem

Robust Arbitrary Order Mixed Finite Element Methods for the Incompressible Stokes Equations with pressure independent velocity errors

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