Two-level defect-correction locally stabilized finite element method for the steady Navier–Stokes equations

This paper proposes a two-level defect-correction stabilized finite element method for the steady Navier–Stokes equations based on local Gauss integration. The method combines the two-level strategy with the defect-correction method under the assumption of the uniqueness condition. Both the simplified and the Newton scheme are proposed and analyzed. Moreover, the numerical illustrations agree completely with the theoretical expectations.     

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.     

A defect-correction stabilized finite element method for Navier–Stokes equations with friction boundary conditions

In this paper, we consider a defect-correction stabilized finite element method for incompressible Navier–Stokes equations with friction boundary conditions whose variational formulation is the variational inequality problem of the second kind with Navier–Stokes operator. In the defect step, an artificial viscosity parameter σ is added to the Reynolds number as a stability factor, and the Oseen iterative scheme is applied in the correction step. $H^1\times L^2$ error estimations are derived for the one-step defect-correction stabilized finite element method. In the end, some numerical results are presented to verify the theoretical analysis.     

A finite element,filtered eddy-viscosity method for the Navier–Stokes equations with large Reynolds number

The direct numerical simulation of the Navier–Stokes system in turbulent regimes is a formidable task due to the disparate scales that have to be resolved. Turbulence modeling attempts to mitigate this situation by somehow accounting for the effects of small-scale behavior on that at large-scales, without explicitly resolving the small scales. One such approach is to add viscosity to the problem; the Smagorinsky and Ladyzhenskaya models and other eddy-viscosity models are examples of this approach. Unfortunately, this approach usually results in over-dampening at the large scales, i.e., large-scale structures are unphysically smeared out. To overcome this fault of simple eddy-viscosity modeling, filtered eddy-viscosity methods that add artificial viscosity only to the high-frequency modes were developed in the context of spectral methods. We apply the filtered eddy-viscosity idea to finite element methods based on hierarchical basis functions. We prove the existence and uniqueness of the finite element approximation and its convergence to solutions of the Navier–Stokes system; we also derive error estimates for finite element approximations.     

The finite volume method based on stabilized finite element for the stationary Navier–Stokes problem

A finite volume method based on stabilized finite element for the two-dimensional stationary Navier–Stokes equations is investigated in this work. A macroelement condition is introduced for constructing the local stabilized formulation for the problem. We obtain the well-posedness of the FVM based on stabilized finite element for the stationary Navier–Stokes equations. Moreover, for quadrilateral and triangular partition, the optimal H¹$H^1$ error estimate of the finite volume solution _uh$u_h$ and L²$L^2$ error estimate for _ph$p_h$ are introduced. Finally, we provide a numerical example to confirm the efficiency of the FVM.     

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.     

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 Crank–Nicolson finite spectral element method and numerical simulations for 2D non-stationary Navier–Stokes equations

In this paper, we first build a semi-discretized Crank–Nicolson (CN) model about time for the two-dimensional (2D) non-stationary Navier–Stokes equations about vorticity–stream functions and discuss the existence, stability, and convergence of the time semi-discretized CN solutions. And then, we build a fully discretized finite spectral element CN (FSECN) model based on the bilinear trigonometric basic functions on quadrilateral elements for the 2D non-stationary Navier–Stokes equations about the vorticity–stream functions and discuss the existence, stability, and convergence of the FSECN solutions. Finally, we utilize two sets of numerical experiments to check out the correctness of theoretical consequences.     

Convergence analysis of an implicit fractional-step method for the incompressible Navier–Stokes equations

In this paper, an implicit fractional-step method for numerical solutions of the incompressible Navier–Stokes equations is studied. The time advancement is decomposed into a sequence of two steps, and the first step can be seen as a linear elliptic problem; on the other hand, the second step has the structure of the Stokes problem. The two problems satisfy the full homogeneous Dirichlet boundary conditions on the velocity. At the same time, we introduce a diffusion term −θΔu in all steps of the schemes. It allows to calculate by the large time step and enhance numerical stability by choosing the proper parameter values of θ. The convergence analysis and error estimates for the intermediate velocities, the end-of step velocities and the pressure solution are derived. Finally, numerical experiments show that the feasibility and effectiveness of this method.     

WEB-spline based mesh-free finite element analysis for the approximation of the stationary Navier–Stokes problem

The objective in this paper is to discuss the existence and the uniqueness of a weighted extended B$B$-spline (WEB-spline) based discrete solution for the stationary incompressible Navier–Stokes equations. The WEB-spline discretization is newly developed methodology which satisfies the inf–sup condition or Ladyshenskaya–Babus?ka–Brezzi (LBB) condition. The main advantage of these new elements over standard finite elements is that they use regular grids instead of irregular partitions of the domain, thus eliminating the difficult and time-consuming pre-processing step. An error estimate for this WEB-spline based discrete solution is also obtained.     

Finite element discretization of the Navier–Stokes equations with a free capillary surface

Summary. The instationary Navier–Stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions. A stable finite element discretization is presented. The key idea is the treatment of the curvature terms by a variational formulation. In the context of a discontinuous in time space–time element discretization stability in (weak) energy norms can be proved. Numerical examples in 2 and 3 space dimensions are given. Received March 26, 1999 / Revised version received October 12, 1999 / Published online November 8, 2000     

Pure Lagrangian and semi-Lagrangian finite element methods for the numerical solution of Navier–Stokes equations

In this paper we propose a unified formulation to introduce Lagrangian and semi-Lagrangian velocity and displacement methods for solving the Navier–Stokes equations. This formulation allows us to state classical and new numerical methods. Several examples are given. We combine them with finite element methods for spatial discretization. In particular, we propose two new second-order characteristics methods in terms of the displacement, one semi-Lagrangian and the other one pure Lagrangian. The pure Lagrangian displacement methods are useful for solving free surface problems and fluid-structure interaction problems because the computational domain is independent of the time and fluid–solid coupling at the interphase is straightforward. However, for moderate to high-Reynolds number flows, they can lead to high distortion in the mesh elements. When this happens it is necessary to remesh and reinitialize the transformation to the identity. In order to assess the performance of the obtained numerical methods, we solve different problems in two space dimensions. In particular, numerical results for a sloshing problem in a rectangular tank and the flow in a driven cavity are presented.     

Analysis of the Yosida method for the incompressible Navier–Stokes equations

The Yosida method was introduced in (Quarteroni et al., to appear) for the numerical approximation of the incompressible unsteady Navier–Stokes equations. From the algebraic viewpoint, it can be regarded as an inexact factorization of the matrix arising from the space and time discretization of the problem. However, its differential interpretation resides on an elliptic stabilization of the continuity equation through the Yosida regularization of the Laplacian (see (Brezis, 1983, Ciarlet and Lions, 1991)). The motivation of this method as well as an extensive numerical validation were given in (Quarteroni et al., to appear).In this paper we carry out the analysis of this scheme. In particular, we consider a first-order time advancing unsplit method. In the case of the Stokes problem, we prove unconditional stability and moreover that the splitting error introduced by the Yosida scheme does not affect the overall accuracy of the solution, which remains linear with respect to the time step. Some numerical experiments, for both the Stokes and Navier–Stokes equations, are presented in order to substantiate our theoretical results.     

The analysis of a finite element method for the three-species Lotka–Volterra competition-diffusion with Dirichlet boundary conditions

A Galerkin finite element method is developed for the two dimensional/three dimensional nonlinear time-dependent three-species Lotka–Volterra competition-diffusion equations on a bounded domain. The existence and uniqueness of the solution to the numerical formulation are proved. An error estimate for the numerical solution is obtained. Numerical computations are carried out to examine the expected orders of accuracy in the error estimates.     

Two-level defect-correction Oseen iterative stabilized finite element methods for the stationary Navier–Stokes equations

Two-level defect-correction Oseen iterative stabilized finite element methods for the stationary Navier–Stokes equations based on local Gauss integration are considered in this paper. The methods combine the defect-correction method and the two-level strategy with the locally stabilized method. Moreover, the stability and convergence of the presented methods are deduced. Finally, numerical tests confirm the theoretical results of the presented methods.     

The convergence of a new parallel algorithm for the Navier–Stokes equations

This paper deals with the convergence and stability of a new parallel algorithm and the error estimates for a particular case of the new parallel algorithm, which is used to solve the incompressible nonstationary Navier–Stokes equations. The theoretical results show that the scheme is (at least) conditionally stable and convergent.     

On the penalty-projection method for the Navier–Stokes equations with the MAC mesh

We deal with the time-dependent Navier–Stokes equations (NSE) with Dirichlet boundary conditions on the whole domain or, on a part of the domain and open boundary conditions on the other part. It is shown numerically that combining the penalty-projection method with spatial discretization by the Marker And Cell scheme (MAC) yields reasonably good results for solving the above-mentioned problem. The scheme which has been introduced combines the backward difference formula of second-order (BDF2, namely Gear’s scheme) for the temporal approximation, the second-order Richardson extrapolation for the nonlinear term, and the penalty-projection to split the velocity and pressure unknowns. Similarly to the results obtained for other projection methods, we estimate the errors for the velocity and pressure in adequate norms via the energy method.