On conforming finite element methods for the inhomogeneous stationary Navier-Stokes equations

Summary We consider the stationary Navier-Stokes equations, written in terms of the primitive variables, in the case where both the partial differential equations and boundary conditions are inhomogeneous. Under certain conditions on the data, the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. A conforming finite element method is presented and optimal estimates for the error of the approximate solution are proved. In addition, the convergence properties of iterative methods for the solution of the discrete nonlinear algebraic systems resulting from the finite element algorithm are given. Numerical examples, using an efficient choice of finite element spaces, are also provided.Supported, in part, by the U.S. Air Force Office of Scientific Research under Grant No. AF-AFOSR-80-0083Supported, in part, by the same agency under Grant No. AF-AFOSR-80-0176-A. Both authors were also partially supported by NASA Contract No. NAS1-15810 while they were in residence at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA 23665, USA     

Kolmogorov equations for randomly perturbed generalized Newtonian fluids

We consider incompressible generalized Newtonian fluids in two space dimensions perturbed by an additive Gaussian noise. The velocity field of such a fluid obeys a stochastic partial differential equation with fully nonlinear drift due to the dependence of viscosity on the shear rate. In particular, we assume that the extra stress tensor is of power law type, i.e. a polynomial of degree , , i.e. the shear thinning case. We prove that the associated Kolmogorov operator K admits at least one infinitesimally invariant measure μ satisfying certain exponential moment estimates. Moreover, K is L²‐unique w.r.t. μ provided , where is the second root of , approximately .     

Least-squares finite element methods for the elasticity problem

A new first-order system formulation for the linear elasticity problem in displacement-stress form is proposed. The formulation is derived by introducing additional variables of derivatives of the displacements, whose combinations represent the usual stresses. Standard and weighted least-squares finite element methods are then applied to this extended system. These methods offer certain advantages such as that they need not satisfy the inf-sup condition which is required in the mixed finite element formulation, that a single continuous piecewise polynomial space can be used for the approximation of all the unknowns, that the resulting algebraic systems are symmetric and positive definite, and that accurate approximations of the displacements and the stresses can be obtained simultaneously. With displacement boundary conditions, it is shown that both methods achieve optimal rates of convergence in the H¹-norm and in the L²-norm for all the unknowns. Numerical experiments with various Poisson ratios are given to demonstrate the theoretical error estimates.     

Attractors for equations describing the flow of generalized Newtonian fluids

Initial boundary-value problems for the equations

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.     

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.     

Local and parallel finite element methods based on two-grid discretizations for the nonstationary Navier-Stokes equations

Numerical Algorithms - In this paper, some local and parallel finite element methods based on two-grid discretizations are proposed and investigated for the unsteady Navier-Stokes equations. The...     

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.     

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.     

Compressible generalized Newtonian fluids

We systematically derive models that would be suitable to describe flows of compressible fluids with the material moduli depending on the symmetric part of the velocity gradient and temperature, within the context of a thermodynamic framework that has been quite successful in developing models to describe the response of bodies that produce entropy while undergoing processes.     

Non-homogeneous generalized Newtonian fluids

We show the existence of weak solutions to the system describing the motion of incompressible, non-homogeneous generalized Newtonian fluids if the extra stress tensor S(ρ, D) possesses p-structure with and variable viscosity. The limiting process in the equation of motion is justified by a variational argument, which is new in this context.     

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.     

Combined finite element and spectral approximation of the Navier-Stokes equations

Summary We present a method for the numerical approximation of Navier-Stokes equations with one direction of periodicity. In this direction a Fourier pseudospectral method is used, in the two others a standard F.E.M. is applied. We prove optimal rate of convergence where the two parameters of discretization intervene independently.

---

Approximation des équations de Navier-Stokes par une méthode éléments finis-spectrale Fourier

Resumé On présente une méthode d'approximation numérique des équations de Navier-Stokes possédant une direction de périodicité. Dans cette direction une méthode pseudospectrale basée sur des développements en série de Fourier est utilisée, dans les deux autres on applique une méthode d'éléments finis standard. On montre que la convergence est optimale et que les deux paramètres de discrétisation peuvent être choisis de façon indépendante.

     

Adaptive finite element methods for Boussinesq equations

An a posteriori error analysis for Boussinesq equations is derived in this article. Then we compare this new estimate with a previous one developed for a regularized version of Boussinesq equations in a previous work. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 214–236, 2000     

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.     

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 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 expanded mixed finite element methods for the generalized forchheimer flows of slightly compressible fluids

The nonlinear Forchheimer equations are used to describe the dynamics of fluid flows in porous media when Darcy's law is not applicable. In this article, we consider the generalized Forchheimer flows for slightly compressible fluids, and then study the expanded mixed finite element method applied to the initial boundary value problem for the resulting degenerate parabolic equation for pressure. The bounds for the solutions, time derivative, and gradient of solutions are established. Utilizing the monotonicity properties of Forchheimer equation and boundedness of solutions, a priori error estimates for solution are obtained in ‐norm, ‐norm as well as for its gradient in ‐norm for all . Optimal ‐error estimates are shown for solutions under some additional regularity assumptions. Numerical results using the lowest order Raviart–Thomas mixed element confirm the theoretical analysis regarding convergence rates. Published 2015. Numer Methods Partial Differential Eq 32: 60–85, 2016     

Weak-strong uniqueness for the generalized Navier-Stokes equations

We give a weak-strong uniqueness result for the weak solutions of the generalized Navier-Stokes equations in Besov space.     

Convergence and stability of finite element nonlinear Galerkin method for the Navier-Stokes equations

In this article, we present a new fully discrete finite element nonlinear Galerkin method, which are well suited to the long time integration of the Navier-Stokes equations. Spatial discretization is based on two-grid finite element technique; time discretization is based on Euler explicit scheme with variable time step size. Moreover, we analyse the boundedness, convergence and stability condition of the finite element nonlinear Galerkin method. Our discussion shows that the time step constraints of the method depend only on the coarse grid parameter and the time step constraints of the finite element Galerkin method depend on the fine grid parameter under the same convergence accuracy. Received February 2, 1994 / Revised version received December 6, 1996