scholarly journals Two-Level Iteration Penalty Methods for the Navier-Stokes Equations with Friction Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Yuan Li ◽  
Rong An
Keyword(s):  
Variational Inequality ◽  
Boundary Conditions ◽  
Variational Inequality Problem ◽  
Stokes Equations ◽  
Mesh Size ◽  
Penalty Methods ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Inequality Problem ◽  
Fine Mesh

This paper presents two-level iteration penalty finite element methods to approximate the solution of the Navier-Stokes equations with friction boundary conditions. The basic idea is to solve the Navier-Stokes type variational inequality problem on a coarse mesh with mesh sizeHin combining with solving a Stokes, Oseen, or linearized Navier-Stokes type variational inequality problem for Stokes, Oseen, or Newton iteration on a fine mesh with mesh sizeh. The error estimate obtained in this paper shows that ifH,h, andεcan be chosen appropriately, then these two-level iteration penalty methods are of the same convergence orders as the usual one-level iteration penalty method.

Two-Level Defect-Correction Method for Steady Navier-Stokes Problem with Friction Boundary Conditions

2016 ◽  
Vol 8 (6) ◽  
pp. 932-952
Author(s):  
An Liu ◽  
Yuan Li ◽  
Rong An
Keyword(s):  
Variational Inequality ◽  
Boundary Conditions ◽  
Variational Inequality Problem ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Correction Method ◽  
Navier Stokes ◽  
Defect Correction ◽  
Inequality Problem ◽  
Fine Mesh

AbstractIn this paper, we present two-level defect-correction finite element method for steady Navier-Stokes equations at high Reynolds number with the friction boundary conditions, which results in a variational inequality problem of the second kind. Based on Taylor-Hood element, we solve a variational inequality problem of Navier-Stokes type on the coarse mesh and solve a variational inequality problem of Navier-Stokes type corresponding to Newton linearization on the fine mesh. The error estimates for the velocity in the H1 norm and the pressure in the L2 norm are derived. Finally, the numerical results are provided to confirm our theoretical analysis.

Two-Level Newton Iteration Methods for Navier-Stokes Type Variational Inequality Problem

2013 ◽  
Vol 5 (1) ◽  
pp. 36-54 ◽  
Author(s):  
Rong An ◽  
Hailong Qiu
Keyword(s):  
Variational Inequality ◽  
Variational Inequality Problem ◽  
Stokes Problem ◽  
Mesh Size ◽  
Newton Iteration ◽  
Navier Stokes ◽  
Element Approximation ◽  
Stabilized Method ◽  
Inequality Problem ◽  
Fine Mesh

AbstractThis paper deals with the two-level Newton iteration method based on the pressure projection stabilized finite element approximation to solve the numerical solution of the Navier-Stokes type variational inequality problem. We solve a small Navier-Stokes problem on the coarse mesh with mesh size H and solve a large linearized Navier-Stokes problem on the fine mesh with mesh size h. The error estimates derived show that if we choose h = O (|logh|1/2H3), then the two-level method we provide has the same H1 and L2 convergence orders of the velocity and the pressure as the one-level stabilized method. However, the L2 convergence order of the velocity is not consistent with that of one-level stabilized method. Finally, we give the numerical results to support the theoretical analysis.

Computation of Inviscid Incompressible Flow Using the Primitive Variable Formulation

1986 ◽  
Vol 108 (1) ◽  
pp. 68-75 ◽  
Author(s):  
S. Abdallah ◽  
H. G. Smith
Keyword(s):  
Boundary Conditions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Type Equation ◽  
Momentum Equation ◽  
Navier Stokes ◽  
Fine Grid ◽  
Navier Stokes Equations ◽  
Pressure Poisson Equation ◽  
Fine Mesh

The primitive variable formulation originally developed for the incompressible Navier–Stokes equations is applied for the solution of the incompressible Euler equations. The unsteady momentum equation is solved for the velocity field and the continuity equation is satisfied indirectly in a Poisson-type equation for the pressure (divergence of the momentum equation). Solutions for the pressure Poisson equation with derivative boundary conditions exist only if a compatibility condition is satisfied (Green’s theorem). This condition is not automatically satisfied on nonstaggered grids. A new method for the solution of the pressure equation with derivative boundary conditions on a nonstaggered grid [25] is used here for the calculation of the pressure. Three-dimensional solutions for the inviscid rotational flow in a 90 deg curved duct are obtained on a very fine mesh (17 × 17 × 29). The use of a fine grid mesh allows for the accurate prediction of the development of the secondary flow. The computed results are in good agreement with the experimental data of Joy [15].

Two-Level Stabilized Finite Volume Methods for the Stationary Navier-Stokes Equations

2013 ◽  
Vol 5 (1) ◽  
pp. 19-35 ◽  
Author(s):  
Tong Zhang ◽  
Shunwei Xu
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Mesh Size ◽  
Finite Volume Methods ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Fine Mesh ◽  
Volume Method

AbstractIn this work, two-level stabilized finite volume formulations for the 2D steady Navier-Stokes equations are considered. These methods are based on the local Gauss integration technique and the lowest equal-order finite element pair. Moreover, the two-level stabilized finite volume methods involve solving one small Navier-Stokes problem on a coarse mesh with mesh size H, a large general Stokes problem for the Simple and Oseen two-level stabilized finite volume methods on the fine mesh with mesh size or a large general Stokes equations for the Newton two-level stabilized finite volume method on a fine mesh with mesh size . These methods we studied provide an approximate solution with the convergence rate of same order as the standard stabilized finite volume method, which involve solving one large nonlinear problem on a fine mesh with mesh size h. Hence, our methods can save a large amount of computational time.

scholarly journals A SUBGRID STABILIZED METHOD FOR NAVIER-STOKES EQUATIONS WITH NONLINEAR SLIP BOUNDARY CONDITIONS

2021 ◽  
Vol 26 (4) ◽  
pp. 528-547
Author(s):  
Xiaoxia Dai ◽  
Chengwei Zhang
Keyword(s):  
Boundary Conditions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Fine Mesh ◽  
High Reynolds ◽  
The Stability ◽  
Nonlinear Slip Boundary Conditions

In this paper, we consider a subgrid stabilized Oseen iterative method for the Navier-Stokes equations with nonlinear slip boundary conditions and high Reynolds number. We provide one-level and two-level schemes based on this stability algorithm. The two-level schemes involve solving a subgrid stabilized nonlinear coarse mesh inequality system by applying m Oseen iterations, and a standard one-step Newton linearization problems without stabilization on the fine mesh. We analyze the stability of the proposed algorithm and provide error estimates and parameter scalings. Numerical examples are given to confirm our theoretical findings.

Parallel iterative finite-element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Kangrui Zhou ◽  
Yueqiang Shang
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Parallel Algorithms ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Parallel Iterative ◽  
Nonlinear Slip Boundary Conditions

AbstractBased on full domain partition, three parallel iterative finite-element algorithms are proposed and analyzed for the Navier–Stokes equations with nonlinear slip boundary conditions. Since the nonlinear slip boundary conditions include the subdifferential property, the variational formulation of these equations is variational inequalities of the second kind. In these parallel algorithms, each subproblem is defined on a global composite mesh that is fine with size h on its subdomain and coarse with size H (H ≫ h) far away from the subdomain, and then we can solve it in parallel with other subproblems by using an existing sequential solver without extensive recoding. All of the subproblems are nonlinear and are independently solved by three kinds of iterative methods. Compared with the corresponding serial iterative finite-element algorithms, the parallel algorithms proposed in this paper can yield an approximate solution with a comparable accuracy and a substantial decrease in computational time. Contributions of this paper are as follows: (1) new parallel algorithms based on full domain partition are proposed for the Navier–Stokes equations with nonlinear slip boundary conditions; (2) nonlinear iterative methods are studied in the parallel algorithms; (3) new theoretical results about the stability, convergence and error estimates of the developed algorithms are obtained; (4) some numerical results are given to illustrate the promise of the developed algorithms.

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