A simple construction of a Fortin operator for the two dimensional Taylor–Hood element

Chen, Long

doi:10.1016/j.camwa.2014.09.003

Cited by 13 publications

(16 citation statements)

References 17 publications

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“…Theorem 3.2 (A priori estimate for pressure-robust finite element methods). For the solution u h of (11) with a reconstruction operator Π that satisfies (12), it holds…”

Section: 2mentioning

confidence: 99%

“…‚ The term η cons,1 pσq " }ν∆ T pdivσq˝p1´Πq} V ‹ 0,h only appears for Π ‰ 1 as in the novel pressurerobust methods and equals the consistency error (13) for σ " ∇u h . ‚ Recall that η cons,2 pqq " 0 if Π satisfies (12) or if q P Q h and Π " 1.…”

Section: (Limits Of) Standard a Posteriori Residual-based Error Boundsmentioning

confidence: 99%

“…The proof of (e) is straight forward and employs integration by parts and the orthogonality of divpv h q onto all q h P Q h if Π " 1 does not satisfy (12). Otherwise, if Π satisfies (12), the assertion follows from divpΠv h q " 0. shows the pressure-dependence also in the efficiency estimate. The volume term η vol pq, ∇u h q scales with the term ν´1}p´q} L 2 .…”

Section: (Limits Of) Standard a Posteriori Residual-based Error Boundsmentioning

confidence: 99%

“…Given a Fortin interpolator I and a reconstruction operator Π with (12) (possibly Π " 1 for divergence-free finite element methods like the Scott-Vogelius finite element method), the novel approach exploits that ΠIv for some divergence-free function v P V 0 is again a divergence-free function in L 2 σ pΩq. Our analysis needs the following assumption on the two operators additional to (12) and (17)- (18). Theorem 5.2 (Novel error estimator for pressure-robust methods).…”

Section: Refined Residual-based Error Boundsmentioning

confidence: 99%

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Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods

2019

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Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness.The main difficulty lies in the volume contribution of the standard residual-based approach that includes the L 2 -norm of the right-hand side. However, the velocity is only steered by the divergencefree part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error.To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the Navier-Stokes equations. The novel error estimators only take the curl of the right-hand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressuredominant situations. This is also confirmed by some numerical examples with the novel pressurerobust modifications of the Taylor-Hood and mini finite element methods.incompressible Navier-Stokes equations and mixed finite elements and pressure robustness and a posteriori error estimators and adaptive mesh refinement

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“…Theorem 3.2 (A priori estimate for pressure-robust finite element methods). For the solution u h of (11) with a reconstruction operator Π that satisfies (12), it holds…”

Section: 2mentioning

confidence: 99%

Section: (Limits Of) Standard a Posteriori Residual-based Error Boundsmentioning

confidence: 99%

Section: (Limits Of) Standard a Posteriori Residual-based Error Boundsmentioning

confidence: 99%

Section: Refined Residual-based Error Boundsmentioning

confidence: 99%

See 2 more Smart Citations

Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods

2019

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“…In this paper, we shall study a family of Taylor–Hood elements and a family of Fortin elements. As is well‐known, the Fortin element is one of the oldest inf‐sup stable and convergent VDB‐Stokes elements while the Taylor–Hood element , a well‐known inf‐sup stable and convergent VDB‐Stokes element, has been popular even until today for numerical solutions of partial differential equations arising from fluid and solid mechanics, see , and so on. On the other hand, Fact strongly supports that both elements are also suitable for singular solution of the PDB‐Stokes problem.…”

Section: Introductionmentioning

confidence: 99%

Staggered Taylor–Hood and Fortin elements for Stokes equations of pressure boundary conditions in Lipschitz domain

Duan

Liu

2019

Numerical Methods Partial

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The purpose of this paper is to develop a general theory on how the inf‐sup stable and convergent elements of the velocity Dirichlet boundary (VDB)‐Stokes problem with no‐slip VDB are still inf‐sup stable and convergent for the pressure Dirichlet boundary (PDB)‐Stokes problem with PDB in Lipschitz domain. The PDB‐Stokes problem in a Lipschitz domain usually only has a singular velocity solution which does not belong to (H1(Ω))2, sharply in contrast to the VDB‐Stokes problem whose velocity solution still belongs to (H1(Ω))2, and unexpectedly, some well‐known inf‐sup stable and convergent VDB‐Stokes elements may or may no longer correctly converge. It turns out that the inf‐sup condition of the PDB‐Stokes problem in Lipschitz domain relies on an unusual variational problem and requires adequate degrees of freedom on the domain boundary. In this paper we propose two families of staggered elements: staggered Taylor–Hood elements ()scriptCPℓ+22−scriptPnormalℓ with ℓ ≥ 1 (continuous in both velocity and pressure) and staggered Fortin elements ()scriptCPm+22−scriptPmdisc with m ≥ 1 (continuous in velocity and discontinuous in pressure) on triangles, for solving the PDB‐Stokes problem in Lipschitz domain. We show that the two families are inf‐sup stable and are correctly convergent for the non‐H1 singular velocity. Numerical results illustrate the proposed elements and the theoretical results.

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Fortin operator for the Taylor–Hood element

2021

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We design a local Fortin operator for the lowest-order Taylor–Hood element in any dimension, which was previously constructed only in 2D. In the construction we use tangential edge bubble functions for the divergence correcting operator. This naturally leads to an alternative inf-sup stable reduced finite element pair. Furthermore, we provide a counterexample to the inf-sup stability and hence to existence of a Fortin operator for the $$P_2$$ P 2 –$$P_0$$ P 0 and the augmented Taylor–Hood element in 3D.

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A simple construction of a Fortin operator for the two dimensional Taylor–Hood element

Cited by 13 publications

References 17 publications

Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods

Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods

Staggered Taylor–Hood and Fortin elements for Stokes equations of pressure boundary conditions in Lipschitz domain

Fortin operator for the Taylor–Hood element

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