A parallel two-level finite element variational multiscale method for the Navier–Stokes equations

A combination method of two-grid discretization approach with a recent finite element variational multiscale algorithm for simulation of the incompressible Navier–Stokes equations is proposed and analyzed. The method consists of a global small-scale nonlinear Navier–Stokes problem on a coarse grid and local linearized residual problems in overlapped fine grid subdomains, where the numerical form of the Navier–Stokes equations on the coarse grid is stabilized by a stabilization term based on two local Gauss integrations at element level and defined by the difference between a consistent and an under-integrated matrix involving the gradient of velocity. By the technical tool of local a priori estimate for the finite element solution, error bounds of the discrete solution are estimated. Algorithmic parameter scalings are derived. Numerical tests are also given to verify the theoretical predictions and demonstrate the effectiveness of the method.     

Two-grid stabilized algorithms for the steady Navier–Stokes equations with damping

This paper presents and studies three two-grid stabilized quadratic equal-order finite element algorithms based on two local Gauss integrations for the steady Navier–Stokes equations with damping. In these algorithms, we first solve a stabilized nonlinear problem on a coarse grid, and then pass the coarse grid solution to a fine grid and solve a stabilized linear problem. Using some nonlinear analysis techniques, we analyze stability of the algorithms and derive optimal order error estimates of the approximate solutions. Theoretical and numerical results show that, when the algorithmic parameters are chosen appropriately, the accuracy of the approximate solutions computed by our two-grid stabilized algorithms is comparable to that of solving a fully stabilized nonlinear problem on the same fine grid; however, our two-grid algorithms save a large amount of CPU time than the one-grid stabilized algorithm.     

A two‐level finite element method for the stationary Navier‐Stokes equations based on a stabilized local projection

In this article, we propose a two‐level finite element method to analyze the approximate solutions of the stationary Navier‐Stokes equations based on a stabilized local projection. The local projection allows to circumvent the Babuska‐Brezzi condition by using equal‐order finite element pairs. The local projection can be used to stabilize high equal‐order finite element pairs. The proposed method combines the local projection stabilization method and the two‐level method under the assumption of the uniqueness condition. The two‐level method consists of solving a nonlinear equation on the coarse mesh and solving a linear equation on fine mesh. The nonlinear equation is solved by the one‐step Newtonian iteration method. In the rest of this article, we show the error analysis of the lowest equal‐order finite element pair and provide convergence rate of approximate solutions. Furthermore, the numerical illustrations coincide with the theoretical analysis expectations. From the view of computational time, the results show that the two‐level method is effective to solve the stationary Navier‐Stokes equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Numerical analysis of a Picard multilevel stabilization of mixed finite volume method for the 2D/3D incompressible flow with large data

In this article, we develop a branch of nonsingular solutions of a Picard multilevel stabilization of mixed finite volume method for the 2D/3D stationary Navier‐Stokes equations without relying on the unique solution condition. The method presented consists of capturing almost all information of initial problem (the nonlinear problems) on the coarsest mesh and then performs one Picard defect correction (the linear problems) on each subsequent mesh based on previous information thus only solving one large linear systems. What is more, the method presented can results in a better coefficient matrix in the model presented with small viscosity. Theoretical results show that the method presented is derived with the convergence rate of the same order as the corresponding finite volume method/finite element method solving the stationary Navier‐Stokes equations on a fine mesh. Therefore, the method presented is definitely more efficient than the standard finite volume method/finite element method. Finally, numerical experiments clearly show the efficiency of the method presented for solving the stationary Navier‐Stokes equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 30–50, 2018     

Two‐level Newton iterative method for the 2D/3D steady Navier‐Stokes equations

A combination method of the Newton iteration and two‐level finite element algorithm is applied for solving numerically the steady Navier‐Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the m Newton iterations for solving the Navier‐Stokes problem on a coarse grid and computing the Stokes problem on a fine grid. Then, the uniform stability and convergence with respect to ν of the two‐level Newton iterative solution are analyzed for the large m and small H and h << H. Finally, some numerical tests are made to demonstrate the effectiveness of the method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

A two‐level variational multiscale method for incompressible flows based on two local Gauss integrations

In this article, a two‐level variational multiscale method for incompressible flows based on two local Gauss integrations is presented. We solve the Navier–Stokes problem on a coarse mesh using finite element variational multiscale method based on two local Gauss integrations, then seek a fine grid solution by solving a linearized problem on a fine grid. In computation, we use the two local Gauss integrations to replace the projection operator without adding any variables. Stability analysis is performed, and error estimates of the method are derived. Finally, a series of numerical experiments are also given, which confirm the theoretical analysis and demonstrate the efficiency of the new method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A three-step Oseen-linearized finite element method for incompressible flows with damping

This paper introduces a three-step Oseen-linearized finite element method for the 2D/3D steady incompressible Navier–Stokes equations with nonlinear damping term. Within this method, we solve a nonlinear problem over a coarse grid followed by solving two Oseen-linearized problems over a fine grid, which possess the same stiffness matrices with only various right-hand sides. We theoretically analyze the stability of the present method, and derive optimal error estimates of the finite element solutions. We conduct a series of numerical experiments which support the theoretical analysis and test the effectiveness of the proposed method. We demonstrate numerically that there is a significant improvement in the accuracy of the approximate solutions over those for the standard two-level method.     

Internal Optimal Controller Synthesis for Navier–Stokes Equations

Barbu and Triggiani (Indiana Univ. Math. J. 2004; 53:1443–1494) have proposed a solution of the internal feedback stabilization problem of Navier–Stokes equations with no-slip boundary conditions. They have shown that any unstable steady-state solution can be exponentially stabilized by a finite-dimensional feedback controller with support in an arbitrary open subset of positive measure. The finite dimension of the feedback controller is minimal and is related to the largest algebraic multiplicity of the unstable eigenvalues of the linearized equation. The feedback law is obtained as a solution of a linear-quadratic control problem. In this paper, we formulate a practical algorithm implementation of the proposed stabilization approach, based on the finite element method, and demonstrate its applicability and effectiveness using an example involving the stabilization of two-dimensional Navier–Stokes equations.     

A parallel finite element variational multiscale method based on fully overlapping domain decomposition for incompressible flows

Based on fully overlapping domain decomposition and a recent variational multiscale method, a parallel finite element variational multiscale method for convection dominated incompressible flows is proposed and analyzed. In this method, each processor computes a local finite element solution in its own subdomain using a global mesh that is locally refined around its own subdomain, where a stabilization term based on two local Gauss integrations is adopted to stabilize the numerical form of the Navier–Stokes equations. Using the technical tool of local a priori estimate for the finite element solution, error bounds of the discrete solution are estimated. Algorithmic parameter scalings are derived. Numerical tests are also given to verify the theoretical predictions and demonstrate the effectiveness of the method. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 856–875, 2015     

Analysis of newton multilevel stabilized finite volume method for the three‐dimensional stationary Navier‐Stokes equations

This article proposes and analyzes a multilevel stabilized finite volume method(FVM) for the three‐dimensional stationary Navier–Stokes equations approximated by the lowest equal‐order finite element pairs. The method combines the new stabilized FVM with the multilevel discretization under the assumption of the uniqueness condition. The multilevel stabilized FVM consists of solving the nonlinear problem on the coarsest mesh and then performs one Newton correction step on each subsequent mesh thus only solving one large linear systems. The error analysis shows that the multilevel‐stabilized FVM provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the stationary Navier–Stokes equations on a fine mesh for an appropriate choice of mesh widths: h_j ～ h_j‐1², j = 1,…,J. Therefore, the multilevel stabilized FVM is more efficient than the standard one‐level‐stabilized FVM. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Local projection FEM stabilization for the time‐dependent incompressible Navier–Stokes problem

We consider conforming finite element (FE) approximations of the time‐dependent, incompressible Navier–Stokes problem with inf‐sup stable approximation of velocity and pressure. In case of high Reynolds numbers, a local projection stabilization method is considered. In particular, the idea of streamline upwinding is combined with stabilization of the divergence‐free constraint. For the arising nonlinear semidiscrete problem, a stability and convergence analysis is given. Our approach improves some results of a recent paper by Matthies and Tobiska (IMA J. Numer. Anal., to appear) for the linearized model and takes partly advantage of the analysis in Burman and Fernández, Numer. Math. 107 (2007), 39–77 for edge‐stabilized FE approximation of the Navier–Stokes problem. Some numerical experiments complement the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1224–1250, 2015     

Two‐grid variational multiscale algorithms for the stationary incompressible Navier‐Stokes equations with friction boundary conditions

Two‐grid variational multiscale (VMS) algorithms for the incompressible Navier‐Stokes equations with friction boundary conditions are presented in this article. First, one‐grid VMS algorithm is used to solve this problem and some error estimates are derived. Then, two‐grid VMS algorithms are proposed and analyzed. The algorithms consist of nonlinear problem on coarse grid and linearized problem (Stokes problem or Oseen problem) on fine grid. Moreover, the stability and convergence of the present algorithms are established. Finally, Numerical results are shown to confirm the theoretical analysis. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 546–569, 2017     

On the semi-discrete stabilized finite volume method for the transient Navier–Stokes equations

A stabilized finite volume method for solving the transient Navier–Stokes equations is developed and studied in this paper. This method maintains conservation property associated with the Navier–Stokes equations. An error analysis based on the variational formulation of the corresponding finite volume method is first introduced to obtain optimal error estimates for velocity and pressure. This error analysis shows that the present stabilized finite volume method provides an approximate solution with the same convergence rate as that provided by the stabilized linear finite element method for the Navier–Stokes equations under the same regularity assumption on the exact solution and a slightly additional regularity on the source term. The stability and convergence results of the proposed method are also demonstrated by the numerical experiments presented.     

The coupling system of Navier–Stokes equations and elastic Navier–Lame equations in a blood vessel

In this paper, the blood flow problem is considered in a blood vessel, and a coupling system of Navier–Stokes equations and linear elastic equations, Navier–Lame equations, in a cylinder with cylindrical elastic shell is given as the governing equations of the problem. We provide two finite element models to simulating the three-dimensional Navier–Stokes equations in the cylinder while the asymptotic expansion method is used to solving the linearly elastic shell equations. Specifically, in order to discrete the Navier–Stokes equations, the dimensional splitting strategy is constructed under the cylinder coordinate system. The spectral method is adopted along the rotation direction while the finite element method is used along the other directions. By using the above strategy, we get a series of two-dimensional-three-components (2D-3C) fluid problems. By introduce the S-coordinate system in E³ and employ the thickness of blood vessel wall as the expanding parameter, the asymptotic expansion method can be established to approximate the solution of the 3D elastic problem. The interface contact conditions can be treated exactly based on the knowledge of tensor analysis. Finally, numerical test shows that our method is reasonable.     

On a hierarchical error estimator combined with a stabilized method for the Navier–Stokes equations

This work combines two complementary strategies for solving the steady incompressible Navier–Stokes model with a zeroth‐order term, namely, a stabilized finite element method and a mesh–refinement approach based on an error estimator. First, equal order interpolation spaces are adopted to approximate both the velocity and the pressure while stability is recovered within the stabilization approach. Also designed to handle advection dominated flows under zeroth‐order term influence, the stabilized method incorporates a new parameter with a threefold asymptotic behavior. Mesh adaptivity driven by a new hierarchical error estimator and built on the stabilized method is the second ingredient. The estimator construction process circumvents the saturation assumption by using an enhancing space strategy which is shown to be equivalent to the error. Several numerical tests validate the methodology. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

解Stokes特征值问题的一种两水平稳定化有限元方法

基于局部Gauss积分,研究了解Stokes特征值问题的一种两水平稳定化有限元方法．该方法涉及在网格步长为H的粗网格上解一个Stokes特征值问题,在网格步长为h=O(H2)的细网格上解一个Stokes问题．这样使其能够仍旧保持最优的逼近精度,求得的解和一般的稳定化有限元解具有相同的收敛阶,即直接在网格步长为h的细网格上解一个Stokes特征值问题．因此,该方法能够节省大量的计算时间．数值试验验证了理论结果．     

An L2 finite element approximation for the incompressible Navier–Stokes equations

This paper utilizes the Picard method and Newton's method to linearize the stationary incompressible Navier–Stokes equations and then uses an LL* approach, which is a least-squares finite element method applied to the dual problem of the corresponding linear system. The LL* approach provides an L²-approximation to a given problem, which is not typically available with conventional finite element methods for nonlinear second-order partial differential equations. We first show that the proposed combination of linearization scheme and LL* approach provides an L²-approximation to the stationary incompressible Navier–Stokes equations. The validity of L²-approximation is proven through the analysis of the weak problem corresponding to the linearized Navier–Stokes equations. Then, the convergence is analyzed, and numerical results are presented.     

Two grid discretizations with backtracking of the stream function form of the Navier-Stokes equations

We analyze a two grid finite element method with backtracking for the stream function formulation of the stationary Navier—Stokes equations. This two grid method involves solving one small, nonlinear coarse mesh system, one linearized system on the fine mesh and one linear correction problem on the coarse mesh. The algorithm and error analysis are presented.     

Acceleration of two-grid stabilized mixed finite element method for the Stokes eigenvalue problem

This paper provides an accelerated two-grid stabilized mixed finite element scheme for the Stokes eigenvalue problem based on the pressure projection. With the scheme, the solution of the Stokes eigenvalue problem on a fine grid is reduced to the solution of the Stokes eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. By solving a slightly different linear problem on the fine grid, the new algorithm significantly improves the theoretical error estimate which allows a much coarser mesh to achieve the same asymptotic convergence rate. Finally, numerical experiments are shown to verify the high efficiency and the theoretical results of the new method.