Analysis of nonconforming stream function and pressure finite element spaces for the Navier-Stokes equations

Cayco, Maria; Nicolaides, R. A.

doi:10.1016/0898-1221(89)90231-9

Cited by 22 publications

(8 citation statements)

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“…The one attraction of stream function formation of the NS equations [14] is that the pressure term can be canceled in the weak form, and there is only one scalar unknown to be solved. The representative work in this field is the analysis of convergence for the standard weak formulation by Cayco and Nicolaides [5,6].Two-grid algorithm is a very promising approach for solving the primitive variable formulation (velocity-pressure formulation) of the NS equations [13,18,20]. Moreover, it can be applied to the stream function formulation of the NS equations [10-12, 21, 26].…”

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A two-grid algorithm based on Newton iteration for the stream function form of the Navier-Stokes equations

Shao

Han

2011

Appl. Math. J. Chin. Univ.

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In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navier-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinear system on the coarse mesh space and two similar linear systems (with same stiffness matrix but different right-hand side) on the fine mesh space.The convergence analysis and error estimation of the algorithm are given for the case of conforming elements. Furthermore, the algorithm produces a numerical solution with the optimal asymptotic H 2 -error. Finally, we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations. §1 Introduction Convergence analysis and error estimation for the finite element method of the primitive variable formulation of the Navier-Stokes(NS) equations have been extensively developed in recent years, e.g. [9,14,15,22,23]. In some research, the stream function formulation is preferred and commonly used for solving the NS problem. The one attraction of stream function formation of the NS equations [14] is that the pressure term can be canceled in the weak form, and there is only one scalar unknown to be solved. The representative work in this field is the analysis of convergence for the standard weak formulation by Cayco and Nicolaides [5,6].Two-grid algorithm is a very promising approach for solving the primitive variable formulation (velocity-pressure formulation) of the NS equations [13,18,20]. Moreover, it can be applied to the stream function formulation of the NS equations [10-12, 21, 26]. The two-grid method is attractive when h H, where H and h are mesh sizes for the coarse grid and fine grid respectively. In the two-grid method, the main point is that the scaling h is how to match H ν for a positive number ν so that the two-grid algorithm produces an approximate solution with Received: 2010-09-25. MR Subject Classification: 65N30, 65N12, 65H99, 65G99.

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A two-grid algorithm based on Newton iteration for the stream function form of the Navier-Stokes equations

Shao

Han

2011

Appl. Math. J. Chin. Univ.

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“…Thus subspace is generally quite difficult to find. Because of this, Cayco and Nicolaides [5] introduced the following equivalent problem:…”

Section: Pressure Recovery Algorithmmentioning

confidence: 99%

“…The standard weak formulation of the stream function version first appeared in 1979 in [10]. In this direction, Cayco and Nicolaides [6,5,4] studied a general analysis of convergence for this standard weak formulation. Once the approximated streamfunction is obtained, the momentum equations can be used to approximate the pressure.…”

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Two Level Finite Element Technique for Pressure Recovery from Stream Function Formulation of the Navier-Stokes Equations

Fairag

2002

Lecture Notes in Computational Science and Engineering

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“…The standard weak formulation of the stream function version first appeared in 1979 in [10]. In this direction, Cayco and Nicolaides [5,4,3] studied a general analysis of convergence for this standard weak formulation.The goal of this paper is to demonstrate that the two-level method can be implemented to approximate solutions for incompressible viscous flow problems with high Reynolds number. …”

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Numerical Computations of Viscous, Incompressible Flow Problems Using a Two-Level Finite Element Method

Fairag

2003

SIAM J. Sci. Comput.

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Abstract. We consider two-level finite element discretization methods for the stream function formulation of the Navier-Stokes equations. The two-level method consists of solving a small nonlinear system on the coarse mesh, then solving a linear system on the fine mesh. The basic result states that the errors between the coarse and fine meshes are related superlinearly. This paper demonstrates that the two-level method can be implemented to approximate efficiently solutions to the Navier-Stokes equations. Two fluid flow calculations are considered to test problems which have a known solution and the driven cavity problem. Stream function contours are displayed showing the main features of the flow. Key words. Two-level method, Navier-Stokes equations, finite element, stream function formulation, Reynolds number AMS subject classifications. 65N35, 76M30, 76D051. Introduction. The numerical treatment of nonlinear problems that arise in areas such as fluid mechanics often requires solving large systems of nonlinear equations. Many methods have been proposed that attempt to solve these systems efficiently; one such class of methods are two-level methods that reduce the computational time. The computational attractions of the methods are that they require the solution of only a small system of nonlinear equations on coarse mesh and one linear system of equations on fine mesh. Apparently, the two-level method was proposed first in [17,16,15] and used for semilinear elliptic problems. The method was implemented for the velocity-pressure formulation of the Navier-Stokes equations in [11,12,13] and for the stream function formulation of the Navier-Stokes equations in [8,18,7].The Navier-Stokes equations may be solved using either the primitive variable or stream function formulation. Here we use the stream function formulation. The attractions of the stream function formulation are that the incompressibility constraint is automatically satisfied, the pressure is not present in the weak form, and there is only one scalar unknown to solve for. The standard weak formulation of the stream function version first appeared in 1979 in [10]. In this direction, Cayco and Nicolaides [5,4,3] studied a general analysis of convergence for this standard weak formulation.The goal of this paper is to demonstrate that the two-level method can be implemented to approximate solutions for incompressible viscous flow problems with high Reynolds number.

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Analysis of nonconforming stream function and pressure finite element spaces for the Navier-Stokes equations

Cited by 22 publications

References 4 publications

A two-grid algorithm based on Newton iteration for the stream function form of the Navier-Stokes equations

A two-grid algorithm based on Newton iteration for the stream function form of the Navier-Stokes equations

Two Level Finite Element Technique for Pressure Recovery from Stream Function Formulation of the Navier-Stokes Equations

Numerical Computations of Viscous, Incompressible Flow Problems Using a Two-Level Finite Element Method

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