Analysis of the iterative penalty method for the Stokes equations

Cheng, Xinbin; Shaikh, Abdul Wasim

doi:10.1016/j.aml.2005.10.021

Cited by 15 publications

(6 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In general we have found the iterative regularization constant to lie in the range of 0.01-0.05. This is also in conformity with recommendations for this constant found in literature (Dai and Cheng, 2008;Lin et al, 2004;Xiao-liang and Shaikh, 2006).…”

Section: Skewed Cavitysupporting

confidence: 93%

“…Iterative penalty thus introduces a regularization step to enforce the incompressibility constraint. We employ the formulation of this method as outlined in (Dai and Cheng, 2008;Lin et al, 2004;Xiao-liang and Shaikh, 2006). The governing equations take the form…”

Section: Incompressible Flow Equations and Penalty Techniquesmentioning

confidence: 99%

“…Momentarily reverting back the notation introduced earlier let us denote (u, p) the solution of equation 1-2 and (uε, pε) the solution to 6-7. It has been shown the following error estimates hold for the weak finite element formulation of the iterative penalty method (Xiao-liang and Shaikh, 2006)…”

Section: Consistencymentioning

confidence: 99%

See 2 more Smart Citations

Augmented Stabilized and Galerkin Least Squares Formulations

Ranjan¹,

Feng²,

Chronopolous

2016

JMR

View full text Add to dashboard Cite

We study incompressible fluid flow problems with stabilized formulations. We introduce an iterative penalty approach to satisfying the divergence free constraint in the Streamline Upwind Petrov Galerkin (SUPG) and Galerkin Least Squares (GLS) formulations, and prove the stability of the formulation. Equal order interpolations for both velocities and pressure variables are utilized for solving problems as opposed to div-stable pairs used earlier. Higher order spectral/hp approximations are utilized for solving two dimensional computational fluid dynamics (CFD) problems with the new formulations named as the Augmented SUPS (ASUPS) and Augmented Galerkin Least Squares (AGLS) formulations. Excellent conservation of mass properties are observed for the problem with open boundaries in confined enclosures. Inexact Newton Krylov methods are used as the non-linear solvers of choice for the problems studied. Faithful representations of all fields of interest are obtained for the problems tested.

show abstract

Section: Skewed Cavitysupporting

confidence: 93%

Section: Incompressible Flow Equations and Penalty Techniquesmentioning

confidence: 99%

See 1 more Smart Citation

Augmented Stabilized and Galerkin Least Squares Formulations

Ranjan¹,

Feng²,

Chronopolous

2016

JMR

View full text Add to dashboard Cite

show abstract

“…Iterated penalty methods for solving Stokes and NSE problem have been used successfully for many years, see e.g. [29][30][31], and analysis of iterated penalty methods is generally done by transforming to a coupled system with artificial incompressibility/pressure regularization. Next, we analyze the methods directly, taking advantage of the divergence-free subspace V h and that the norm of its orthogonal complement in X h is the divergence norm.…”

Section: Connection To the Classical Iterated Penalty Methodsmentioning

confidence: 99%

“…Consider now the classical iterated penalty algorithm, based on a formulation from [29] and in the same spirit as in [33,31,30], but here we use exact integration in the penalty term. Step 0:…”

Section: Algorithm 41 (Pointwise Divergence Free Solution For Couplementioning

confidence: 99%

On reducing the splitting error in Yosida methods for the Navier–Stokes equations with grad-div stabilization

Rebholz

Xiao

2015

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

This paper analyzes the accuracy of the 'discretize-then-split' Yosida solver for incompressible flow problems, when divergencefree elements are used together with grad-div stabilization (with parameter γ ). The Yosida method uses an inexact block LU factorization to create linear algebraic systems that are easier to solve, but at the expense of accuracy. We prove the difference between solutions of the exact and approximated linear algebraic systems is O(γ −2 ) in the natural norms of the associated finite element problems, and thus that full accuracy can be obtained by the Yosida method if large γ is used (γ ≥ 10 is sufficient in our numerical examples). The proof is based on transforming the Yosida inexact linear algebraic system into finite element problems, and analyzing these problems with finite element techniques based on pointwise divergence-free subspaces and their orthogonal complements.

show abstract

Analysis of an iterative penalty method for Navier–Stokes equations with nonlinear slip boundary conditions

Dai

Tang

2012

Numerical Methods in Fluids

View full text Add to dashboard Cite

SUMMARYWhen we use penalty method to solve Navier–Stokes equations with nonlinear slip boundary conditions, the penalty parameter ϵ should be sufficiently small to yield an accurate approximation and the choice of ϵ depends on the mesh size h. However, the condition number of the associated stiffness matrix becomes large if ϵ is too small, which leads to the unstable computation. In this paper, we design an iterative penalty method for the problem and give some error estimates. In our algorithm, we can use a not very small penalty parameter ϵ independent of h to avoid the unstable computation. Numerical examples are given to show that the algorithm is very effective and powerful. Copyright © 2012 John Wiley & Sons, Ltd.

show abstract

Analysis of the iterative penalty method for the Stokes equations

Cited by 15 publications

References 8 publications

Augmented Stabilized and Galerkin Least Squares Formulations

Augmented Stabilized and Galerkin Least Squares Formulations

On reducing the splitting error in Yosida methods for the Navier–Stokes equations with grad-div stabilization

Analysis of an iterative penalty method for Navier–Stokes equations with nonlinear slip boundary conditions

Contact Info

Product

Resources

About