Two-level pressure projection finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions

The two-level pressure projection stabilized finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier-Stokes type variational inequality problem of the second kind. Based on the P₁-P₁ triangular element and using the pressure projection stabilized finite element method, we solve a small Navier-Stokes type variational inequality problem on the coarse mesh with mesh size H and solve a large Stokes type variational inequality problem for simple iteration or a large Oseen type variational inequality problem for Oseen iteration on the fine mesh with mesh size h. The error analysis obtained in this paper shows that if h=O(H²), the two-level stabilized methods have the same convergence orders as the usual one-level stabilized finite element methods, which is only solving a large Navier-Stokes type variational inequality problem on the fine mesh. Finally, numerical results are given to verify the theoretical analysis.     

On Error Estimates of the Projection Methods for the Navier-Stokes Equations: Second-order Schemes

We present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible Navier-Stokes equations. The error analysis is accomplished by interpreting the respective projection schemes as second-order time discretizations of a perturbed system which approximates the Navier-Stokes equations. Numerical results in agreement with the error analysis are also presented.

     

Derivation of projection methods from formal integration of the Navier-Stokes equations

Projection methods constitute a class of numerical methods for solving the incompressible Navier-Stokes equations. These methods operate using a two-step procedure in which the zero-divergence constraint on the velocity is first relaxed while the velocity evolves, then after a certain period of time the resulting velocity field is projected onto a divergence-free subspace. Although these methods can be quite efficient, there have been certain concerns regarding their formulation. In this paper we show how a formal integration of the Navier-Stokes equations leads to a new and general procedure for the derivation of projection methods. By following this procedure, we show how each of three practical projection methods approximates a system of equations that is equivalent to the Navier-Stokes equations. We also show how the auxiliary boundary conditions required in projection methods are related to the physical boundary conditions. These results should allay the concerns regarding the legitimacy of projection methods, and may assist in their future development.     

A new error correction method for the stationary Navier-Stokes equations based on two local Gauss integrations

A new error correction method for the stationary Navier-Stokes equations based on two local Gauss integrations is presented. Applying the orthogonal projection technique, we introduce two local Gauss integrations as a stabilizing term in the error correction method, and derive a new error correction method. In both the coarse solution computation step and the error computation step, a locally stabilizing term based on two local Gauss integrations is introduced. The stability and convergence of the new error correction algorithm are established. Numerical examples are also presented to verify the theoretical analysis and demonstrate the efficiency of the proposed method.     

A fractional spectral method with applications to some singular problems

In this paper we propose and analyze fractional spectral methods for a class of integro-differential equations and fractional differential equations. The proposed methods make new use of the classical fractional polynomials, also known as Müntz polynomials. We first develop a kind of fractional Jacobi polynomials as the approximating space, and derive basic approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces. We then construct efficient fractional spectral methods for some integro-differential equations which can achieve spectral accuracy for solutions with limited regularity. The main novelty of the proposed methods is that the exponential convergence can be attained for any solution u(x) with u(x ^1/λ ) being smooth, where λ is a real number between 0 and 1 and it is supposed that the problem is defined in the interval (0,1). This covers a large number of problems, including integro-differential equations with weakly singular kernels, fractional differential equations, and so on. A detailed convergence analysis is carried out, and several error estimates are established. Finally a series of numerical examples are provided to verify the efficiency of the methods.     

Nonconforming mixed finite element approximation to the stationary Navier-Stokes equations on anisotropic meshes

The main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary Navier-Stokes equations under anisotropic meshes. That is, the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure. The convergence analysis is presented and the optimal error estimates both in a broken H¹-norm for the velocity and in an L²-norm for the pressure are derived on anisotropic meshes.     

Projection and projection-difference methods for the solution of the Navier-Stokes equations

The projection and projection-difference methods for the approximate solution of the nonlinear unsteady Navier-Stokes equations in a bounded two-dimensional region are studied. Asymptotic estimates for the convergence rate of the approximate solutions and the time and space derivatives in the uniform topology are obtained.     

On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations

Summary This paper is a continuation of our previous work [10] on projection methods. We study first existing higher order projection schemes in the semidiscretized form for the Navier-Stokes equations. One error analysis suggests that the precision of these schemes is most likely plagued by the inconsistent Neumann boundary condition satisfied by the pressure approximations. We then propose a penalty-projection scheme for which we obtain improved error estimates.This work is partially supported by NSF grant MS-8802596.     

A NEW STABILIZED FINITE ELEMENT METHOD FOR SOLVING TRANSIENT NAVIER-STOKES EQUATIONS WITH HIGH REYNOLDS NUMBER

In this paper, we present a new stabilized finite element method for transient Navier-Stokes equations with high Reynolds number based on the projection of the velocity and pressure. We use Taylor-Hood elements and the equal order elements in space and second order difference in time to get the fully discrete scheme. The scheme is proven to possess the absolute stability and the optimal error estimates. Numerical experiments show that our method is effective for transient Navier-Stokes equations with high Reynolds number and the results are in good agreement with the value of subgrid-scale eddy viscosity methods, Petro-Galerkin finite element method and streamline diffusion method.     

Convergence analysis of a projection semi-implicit coupling scheme for fluid–structure interaction problems

In this paper, we provide a convergence analysis of a projection semi-implicit scheme for the simulation of fluid–structure systems involving an incompressible viscous fluid. The error analysis is performed on a fully discretized linear coupled problem: a finite element approximation and a semi-implicit time-stepping strategy are respectively used for space and time discretization. The fluid is described by the Stokes equations, the structure by the classical linear elastodynamics equations (linearized elasticity, plate or shell models) and all changes of geometry are neglected. We derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least ${\sqrt{\delta t}}In this paper, we provide a convergence analysis of a projection semi-implicit scheme for the simulation of fluid–structure systems involving an incompressible viscous fluid. The error analysis is performed on a fully discretized linear coupled problem: a finite element approximation and a semi-implicit time-stepping strategy are respectively used for space and time discretization. The fluid is described by the Stokes equations, the structure by the classical linear elastodynamics equations (linearized elasticity, plate or shell models) and all changes of geometry are neglected. We derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least ?{dt}{\sqrt{\delta t}}. Finally, some numerical experiments that confirm the theoretical analysis are presented.     

Analysis of a sequential regularization method for the unsteady Navier-Stokes equations

The incompressibility constraint makes Navier-Stokes equations difficult. A reformulation to a better posed problem is needed before solving it numerically. The sequential regularization method (SRM) is a reformulation which combines the penalty method with a stabilization method in the context of constrained dynamical systems and has the benefit of both methods. In the paper, we study the existence and uniqueness for the solution of the SRM and provide a simple proof of the convergence of the solution of the SRM to the solution of the Navier-Stokes equations. We also give error estimates for the time discretized SRM formulation.

     

Semilocal convergence of an eighth-order method in Banach spaces and its computational efficiency

The aim of this paper is to study the semilocal convergence of the eighth-order iterative method by using the recurrence relations for solving nonlinear equations in Banach spaces. The existence and uniqueness theorem has been proved along with priori error bounds. We have also presented the comparative study of the computational efficiency in case of R^m with some existing methods whose semilocal convergence analysis has been already discussed. Finally, numerical application on nonlinear integral equations is given to show our approach.     

A globally convergent BFGS method for nonlinear monotone equations without any merit functions

Since 1965, there has been significant progress in the theoretical study on quasi-Newton methods for solving nonlinear equations, especially in the local convergence analysis. However, the study on global convergence of quasi-Newton methods is relatively fewer, especially for the BFGS method. To ensure global convergence, some merit function such as the squared norm merit function is typically used. In this paper, we propose an algorithm for solving nonlinear monotone equations, which combines the BFGS method and the hyperplane projection method. We also prove that the proposed BFGS method converges globally if the equation is monotone and Lipschitz continuous without differentiability requirement on the equation, which makes it possible to solve some nonsmooth equations. An attractive property of the proposed method is that its global convergence is independent of any merit function.We also report some numerical results to show efficiency of the proposed method.

     

Equations de navier-stokes 3-D avec force de coriolis et viscosité verticale évanescente

We consider the 3-D Navier-Stokes equations with Coriolis force of order 1/z and vanishing vertical viscosity of order ɛ. For suitable initial data, we prove some global or long-time existence results. Moreover, we obtain convergence as ɛ goes to 0 to the 2-D Navier-Stokes equations. We deal with periodic boundary conditions and non-homogeneous strain: in this case, we compute and justify the corrector.     

Solving nonlinear mixed Volterra–Fredholm integral equations with two dimensional block-pulse functions using direct method

Few numerical methods such as projection methods, time collocation method, trapezoidal Nystrom method, Adomian decomposition method and some else are used for mixed Volterra–Fredholm integral equations. The main purpose of this paper is to use the piecewise constant two-dimensional block-pulse functions (2D-BPFs) and their operational matrices for solving mixed nonlinear Volterra–Fredholm integral equations of the first kind (VFIE). This method leads to a linear system of equations by expanding unknown function as 2D-BPFs with unknown coefficients. The properties of 2D-BPFs are then utilized to evaluate the unknown coefficients. The error analysis and rate of convergence are given. Finally, some numerical examples show the implementation and accuracy of this method.     

Projektive Newton-Verfahren und Anwendungen auf nichtlineare Randwertaufgaben

Summary In this paper we consider the following Newton-like methods for the solution of nonlinear equations. In each step of the Newton method the linear equations are solved approximatively by a projection method. We call this a Projective Newton method. For a fixed projection method the approximations often are the same as those of the Newton method applied to a nonlinear projection method. But the efficiency can be increased by adapting the accuracy of the projection method to the convergence of the approximations. We investigate the convergence and the order of convergence for these methods. The results are applied to some Projective Newton methods for nonlinear two point boundary value problems. Some numerical results indicate the efficiency of these methods.

     

关于非定常不可压Navier-Stokes方程的时间高精度隐式差分方法

The incompressible Navier-Stokes equations,upon spatial discretization,become a system of differential algebraic equations,formally of index2.But due to the special forms of the discrete gradient and disrete divergence,its index can be regarded as 1.Thus,in this paper,a systematic approach following the ODE theory and methods is presented for the construction of high-order time-accurate implicit schemes for the incompressible Navier-Stokes equations,with projection methods for efficiency of numerical solution.The 3rd order 3-step BDF with componentconsistent pressure-correction projection method is a first attempt in this direction;the related iterative solution of the auxiliary velocyty,the boundary conditions and the stability of the algorithm are discussed.Results of numerical tests on the incompressible Navier-Stokes equations with an exact solution are presented,confirming the accureacy,stability and component-consistency of the proposed method.     

A UNIFIED FRAMEWORK FOR SOME INEXACT PROXIMAL POINT ALGORITHMS*

We present a unified framework for the design and convergence analysis of a class of algorithms based on approximate solution of proximal point subproblems. Our development further enhances the constructive approximation approach of the recently proposed hybrid projection–proximal and extragradient–proximal methods. Specifically, we introduce an even more flexible error tolerance criterion, as well as provide a unified view of these two algorithms. Our general method possesses global convergence and local (super)linear rate of convergence under standard assumptions, while using a constructive approximation criterion suitable for a number of specific implementations. For example, we show that close to a regular solution of a monotone system of semismooth equations, two Newton iterations are sufficient to solve the proximal subproblem within the required error tolerance. Such systems of equations arise naturally when reformulating the nonlinear complementarity problem.

     

Assessment of subgrid-scale models for the incompressible Navier-Stokes equations

In this paper, we assess two kinds of subgrid finite element methods for the two-dimensional (2D) incompressible Naver-Stokes equations (NSEs). These methods introduce subgrid-scale (SGS) eddy viscosity terms which do not act on the large flow structures. The eddy viscous terms consist of the fluid flow fluctuation strain rate stress tensors. The fluctuation tensor can be calculated by a elliptic projection or a simple L² projection (projective filter) in finite element spaces. The finite element pair P₂/P₁ is adopted to numerically implement analysis and computation. We give a complete error analysis based on the assumptions of some regularity conditions. On the part of numerical tests, the numerical computations for the stationary flows show that the numerical results agree with some benchmark solutions and theoretical analysis very well. Furthermore, the given SGS models are applied to the non-stationary fluid flows.     

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

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.