A fully discrete stabilized finite element method for the time-dependent Navier-Stokes equations

In this article, we consider a fully discrete stabilized finite element method based on two local Gauss integrations for the two-dimensional time-dependent Navier-Stokes equations. It focuses on the lowest equal-order velocity-pressure pairs. Unlike the other stabilized method, the present approach does not require specification of a stabilization parameter or calculation of higher-order derivatives, and always leads to a symmetric linear system. The Euler semi-implicit scheme is used for the time discretization. It is shown that the proposed fully discrete stabilized finite element method results in the optimal order bounds for the velocity and pressure.     

Stabilized finite element method based on the Crank-Nicolson extrapolation scheme for the time-dependent Navier-Stokes equations

This paper provides an error analysis for the Crank-Nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two-dimensional time-dependent Navier-Stokes problem, where the finite element space pair for the approximation of the velocity and the pressure is constructed by the low-order finite element: the quadrilateral element or the triangle element with mesh size . Error estimates of the numerical solution to the exact solution with are derived.

     

Two-grid stabilized mixed finite element method for fully discrete reaction-diffusion equations

Two-grid mixed finite element method is proposed based on backward Euler schemes for the unsteady reaction-diffusion equations. The scheme combines with the stabilized mixed finite element scheme by using the lowest equal-order pairs for the velocity and pressure. The space two-grid method is also used to reduce the time consuming. The benefits of this approach are to avoid the higher derivative, but to have more favorable stability, and to get the numerical solution of the two unknown variables simultaneously. Stability analysis and error estimates are given in this work. Finally, the theoretical results are verified by the numerical examples.     

A mixed finite element approximation of the Navier-Stokes equations

Summary We study in this paper the convergence of a new mixed finite element approximation of the Navier-Stokes equations. This approximation uses low order Lagrange elements, leads to optimal order of convergence for the velocity and the pressure, and induces an efficient numerical algorithm for the solution of this problem.     

An analysis of a mixed finite element method for the Navier-Stokes equations

Summary A simple mixed finite element method is developed to solve the steady state, incompressible Navier-Stokes equations in a neighborhood of an isolated—but not necessarily unique—solution. Convergence is established under very mild restrictions on the triangulation, and, when the solution is sufficiently smooth, optimal error bounds are obtained.     

定常Navier-Stokes方程的一种高效稳定有限元方法

提出了二维定常Navier-Stokes(N-S)方程的一种两层稳定有限元方法.该方法基于局部高斯积分技术,通过不满足inf-sup条件的低次等阶有限元对N-S方程进行有限元求解.该方法在粗网格上解定常N-S方程,在细网格上只需解一个Stokes方程.误差分析和数值试验都表明:两层稳定有限元方法与直接在细网格上采用的传统有限元方法得到的解具有同阶的收敛性,但两层稳定有限元方法节省了大量的工作时间.     

A stabilized finite element method for the incompressible magnetohydrodynamic equations

Summary. We propose and analyze a stabilized finite element method for the incompressible magnetohydrodynamic equations. The numerical results that we present show a good behavior of our approximation in experiments which are relevant from an industrial viewpoint. We explain in particular in the proof of our convergence theorem why it may be interesting to stabilize the magnetic equation as soon as the hydrodynamic diffusion is small and even if the magnetic diffusion is large. This observation is confirmed by our numerical tests. Received August 31, 1998 / Revised version received June 16, 1999 / Published online June 21, 2000     

Convergence and stability of two-level penalty mixed finite element method for stationary Navier-Stokes equations

The two-level penalty mixed finite element method for the stationary Navier-Stokes equations based on Taylor-Hood element is considered in this paper. Two algorithms are proposed and analyzed. Moreover, the optimal stability analysis and error estimate for these two algorithms are provided. Finally, the numerical tests confirm the theoretical results of the presented algorithms.     

A mixed finite element method for the Stokes equations

A new mixed finite element for the Stokes equations is considered. This new finite element is based on a mixed formulation of the Stokes problem in which the gradient of the velocity is introduced and the velocity is approximated by the Raviart-Thomas element [1]. Optimal error estimates are derived. The number of degrees of freedom, for this element, is the lowest possible, and the local conservation of the mass is assured. A hybrid version of the mixed method is also considered. Finally, some numerical results for the incompressible Navier-Stokes equations are presented. © 1994 John Wiley & Sons, Inc.     

Stabilized multiscale finite element method for the stationary Navier-Stokes equations

In the paper, a stabilized multiscale finite element method for the stationary incompressible Navier-Stokes equations is considered. The method is a Petrov-Galerkin approach based on the multiscale enrichment of the standard polynomial space enriched with the unusual bubble functions which no longer vanish on every element boundary for the velocity space. The stability of the P₁-P₀ triangular element (or the Q₁-P₀ quadrilateral element) is established. And the optimal error estimates of the stabilized multiscale finite element method for the stationary Navier-Stokes equations are obtained.     

Unified analysis for stabilized methods of low-order mixed finite elements for stationary Navier-Stokes equations

A unified analysis is presented for the stabilized methods including the pres-sure projection method and the pressure gradient local projection method of conforming and nonconforming low-order mixed fi...     

Two-level stabilized nonconforming finite element method for the Stokes equations

In this article, we present a new two-level stabilized nonconforming finite elements method for the two dimensional Stokes problem. This method is based on a local Gauss integration technique and the mixed nonconforming finite element of the NCP ₁ — P ₁ pair (nonconforming linear element for the velocity, conforming linear element for the pressure). The two-level stabilized finite element method involves solving a small stabilized Stokes problem on a coarse mesh with mesh size H and a large stabilized Stokes problem on a fine mesh size h = H/3. Numerical results are presented to show the convergence performance of this combined algorithm.     

A combined finite element and Marker and cell method for solving Navier-Stokes equations

Summary A new method is developed to approximate Navier-Stokes equations for incompressible fluids on polygonal domains. This method is based on a simple finite element approximation adapted to the Marker and Cell technique. Its main originality lies in the fact that the components of the velocity are approximated on distinct interlaced networks. The error analysis shows that this method is of order one.     

Stabilized finite element method for Navier-Stokes equations with physical boundary conditions

This paper deals with the numerical approximation of the 2D and 3D Navier-Stokes equations, satisfying nonstandard boundary conditions. This lays on the finite element discretisation of the corresponding Stokes problem, which is achieved through a three-fields stabilized mixed formulation. A priori and a posteriori error bounds are established for the nonlinear problem, ascertaining the convergence of the method. Finally, numerical tests are presented, including mesh refinement via error indicators.

     

Analysis of multiscale finite element method for the stationary Navier-Stokes equations

In this work, a multiscale finite element method is proposed for the stationary incompressible Navier-Stokes equations. And the inf-sup stability of the method for the P₁/P₁ triangular element is established. The optimal error estimates are obtained.     

A posteriori error estimators for a two-level finite element method for the Navier-Stokes equations

Two- and multilevel truncated Newton finite element discretizations are presently a very promising approach for approximating the (nonlinear) Navier-Stokes equations describing the equilibrium flow of a viscous, incompressible fluid. Their combination with mesh adaptivity is considered in this article. Specifically, locally calculable a posteriori error estimators are derived, with full mathematical support, for the basic two-level discretization of the Navier-Stokes equations. © 1996 John Wiley & Sons, Inc.     

On a reduced mixed s-v least-squares finite element formulation for the incompressible Navier-Stokes equations

In the present work a mixed finite element based on a least-squares approach (LSFEM) is proposed. We consider a formulation for Newtonian fluid flow, which is described by the incompressible Navier-Stokes equations. The starting point is a div-grad three-field first-order system with stresses, velocities, and pressure as unknowns. Following the idea in CAI et al. [1], this three-field formulation can be transformed into a reduced stress-velocity (s-v) two-field formulation, which is the basis for the associated minimization problem. In order to show the applicability of the considered approach a numerical example is presented at the end of the paper. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence and stability of a stabilized finite volume method for the stationary Navier-Stokes equations

In this paper, a new stabilized finite volume method is studied and developed for the stationary Navier-Stokes equations. This method is based on a local Gauss integration technique and uses the lowest equal order finite element pair P ₁–P ₁ (linear functions). Stability and convergence of the optimal order in the H ¹-norm for velocity and the L ²-norm for pressure are obtained. A new duality for the Navier-Stokes equations is introduced to establish the convergence of the optimal order in the L ²-norm for velocity. Moreover, superconvergence between the conforming mixed finite element solution and the finite volume solution using the same finite element pair is derived. Numerical results are shown to support the developed convergence theory.