Least-Squares Finite Element Method for the Stokes Problem with Zero Residual of Mass Conservation

Chang, Ching L.; Nelson, John J.

doi:10.1137/s0097539794273368

Cited by 88 publications

(97 citation statements)

References 13 publications

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“…To overcome this we introduce an iterative penalization scheme, on the similar lines of [18,34], for the transient pressure-velocity-stress first-order system of Navier-Stokes equations. By penalty method, we recast the constrained minimization problem into an unconstrained minimization problem through the use of the penalty method [14,44].…”

Section: A Least-squares Finite Element Model For Flows Of Viscous Inmentioning

confidence: 99%

RETRACTED: Higher-order Spectral/hp Finite Element Technology for Shells and Flows of Viscous Incompressible Fluids

Vallala

Reddy

2013

MCA

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Abstract-This study deals with the use of high-order spectral/hp approximation functions in finite element models of various of nonlinear boundary-value and initial-value problems arising in the fields of structural mechanics and flows of viscous incompressible fluids. For many of these classes of problems, the high-order (typically, polynomial order 4  p ) spectral/hp finite element technology offers many computational advantages over traditional low-order (i.e., 3 < p ) finite elements. For instance, higher-order spectral/hp finite element procedures allow us to develop robust structural elements such as beams, plates, and shells in a purely displacement-based setting, which avoid all forms of numerical locking. For fluid flows, when combined with least-squares variational principles, the higher-order spectral/hp technology allows us to develop efficient finite element models that always yield a symmetric positive-definite (SPD) coefficient matrix and, hence, robust iterative solvers can be used. Also, the use of spectral/hp finite element technology results in a better conservation of physical quantities like dilatation, volume, and mass, and stable evolution of variables with time for transient flows. The present study considers the weak-form based displacement finite element models elastic shells and the least-squares finite element models of the Navier-Stokes equations governing flows of viscous incompressible fluids. Numerical solutions of several nontrivial benchmark problems are presented to illustrate the accuracy and robustness of the developed finite element technology.

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Section: A Least-squares Finite Element Model For Flows Of Viscous Inmentioning

confidence: 99%

RETRACTED: Higher-order Spectral/hp Finite Element Technology for Shells and Flows of Viscous Incompressible Fluids

Vallala

Reddy

2013

MCA

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“…In the following we consider a steady Stokes flow around a cylinder, see CHANG AND NELSON [5], in two dimensions for a Reynolds number Re = 100. We use cubic overconstrained s-v RT 1 P 3 elements and standard s-v RT 1 P 3 elements, see [2], for the comparison of the solution of the horizontal velocity v 1 on different meshes.…”

Section: Least-squares Methodsmentioning

confidence: 99%

“…Here, we compare the solution of the horizontal velocity v 1 for uniform meshes with 50 elements per edge with the reference solution of GHIA ET AL. [5]. For the comparison cubic overconstrained s-v RT 1 P 3 elements and standard s-v-p RT 1 P 3 P 1 elements, see [4], are used.…”

Section: Least-squares Methodsmentioning

confidence: 99%

Performance aspects of a mixed s‐v LSFEM for the incompressible Navier‐Stokes equations with improved mass conservation

Schwarz

Schröder

Serdas

et al. 2013

Proc Appl Math and Mech

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In the present contribution we propose an improved mixed least-squares finite element method (LSFEM) and compare it with standard LSFEMs with respect to performance aspects. In detail, we consider an approach for Newtonian fluid flow, which is described by the incompressible Navier-Stokes equations. The basis for the associated symmetric minimization problem is a reduced stress-velocity (s-v) two-field approach, see e.g. CAI ET AL.[1] and SCHWARZ & SCHRÖDER [2]. The main idea for the proposed formulation is to add an additional equation, which yields an overconstrained first-order system. This approach does not introduce additional unknowns, so the advantage of two variables (stresses and velocities) remains. Finally, we present a numerical example in order to show the capability of the proposed formulation.

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“…[2]). A similar idea to weighting the divergence term is to enforce it through a Lagrange multiplier strategy [13]. In this case, the operator is still symmetric, but the now indefinite operator is not effectively solved using a standard multigrid method.…”

Section: Possible Remediesmentioning

confidence: 99%

On Mass‐Conserving Least‐Squares Methods

Heys¹,

Lee²,

Manteuffel³

et al. 2006

SIAM J. Sci. Comput.

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Abstract. Least-squares variational methods have several practical and theoretical advantages for solving elliptic partial differential equations, including symmetric positive definite discrete operators and a sharp error measure. One of the potential drawbacks, especially in three dimensions, is that mass conservation is achieved only in a least-squares sense, and underresolved solutions are especially vulnerable to poor conservation. For the stationary Navier-Stokes equations, which are typically rewritten as a larger system of first-order equations, the loss of mass in the approximate solution is strongly dependent upon the boundary conditions used. A new first-order system reformulation of the Navier-Stokes equations is presented that admits a wider range of mass-conserving boundary conditions. This new formulation is shown to provide both excellent mass conservation and excellent algebraic multigrid performance for three different problems, a square channel, backwardfacing step, and branching tubes with two generations of bifurcations.

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Least-Squares Finite Element Method for the Stokes Problem with Zero Residual of Mass Conservation

Cited by 88 publications

References 13 publications

RETRACTED: Higher-order Spectral/hp Finite Element Technology for Shells and Flows of Viscous Incompressible Fluids

RETRACTED: Higher-order Spectral/hp Finite Element Technology for Shells and Flows of Viscous Incompressible Fluids

Performance aspects of a mixed s‐v LSFEM for the incompressible Navier‐Stokes equations with improved mass conservation

On Mass‐Conserving Least‐Squares Methods

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