非定常Stokes问题的矩形Crouzeix-Raviart型各向异性非协调元变网格方法

该文给出了关于速度-压力型非定常Stokes问题的一个 矩形 Crouzeix-Raviart 型各向异性非协调有限元的变网格逼近格式.并用一些新的技巧和方法导出了各向异性网格下的有关速度和压力的最优误差估计.     

二阶特征值问题的非协调元逼近

本文以非协调三角形线性元为例,讨论了二阶特征值问题的非协调有限元逼近,基于二阶变分问题非协调有限元逼近的有关分析结果,不仅得到了特征值逼近解的误差估计,而且得到了特征函数逼近解的最优的L~2-误差估计和拟最优的L~∞-误差估计。     

解Stokes问题的P1非协调四边形元的稳定化方法

本文研究了用(~P)_1-Q_0元(其中(~P)_1表示P_1非协调四边形元)解Stokes问题的非协调混合有限元稳定化逼近方法.(~P)_1-Q_0元不满足LBB条件(见[7,14] ),因而其不能直接用来求解Stokes问题.受[3] 的启发,我们提出了一种用(~P)_1-Q_0元解Stokes问题的稳定化方法,证明了这种方法的稳定性和离散问题解的存在唯一性,得到了最优误差估计.文章最后给出的数值算例验证了我们的理论结果.     

四阶特征值问题的非协调元逼近

本文讨论了二类典型的四阶特征值问题的非协调有限元逼近;得到了最优的低模误差估计和拟最优的L(?)-误差估计。     

Stokes型积分——微分方程的Galerkin近似

本文讨论一类具有Stokes方程结构的积分一微分方程的Galerkin近似，论证了近似解的存在唯一性，并分别导出速度和压力近似解的最优阶L_2模误差估计。     

Stokes型积分微分方程的质量集中各向异性非协调有限元分析

本文将Crouzeix-Raviart型非协调三角形元应用到发展型Stokes积分微分方程,给出了其质量集中非协调有限元逼近格式.在各向异性网格下,导出了速度的L2模和能量模及压力的L2模的误差估计.     

Navier—Stokes方程的变网格非协调有限元法

本文通过所谓的速度－压力型公式讨论了Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程的变网格非协调有限元逼近，得到了在模意义下的速度，压力误差估计，且在一定条件下，某些误差估计能达到最优。     

Navier－Stokes方程的变网格非协调有限元法

本文通过所谓的速度－压力型公式讨论了Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程的变网格非协调有限元逼近，得到了在确定模意义下的速度、压力误差估计，且在一定条件下，某些误差估计能达到最优。     

Stokes问题非协调混合有限元超收敛分析

本文通过引入全新的技巧,研究了Stokes问题的非协调混合有限元方法,得到了关于速度与压力的超逼近性质.进一步地通过构造一个恰当的插值后处理算子,还得到了关于速度的整体超收敛结果.     

电报方程的类Wilson非协调有限元分析

本文在矩形网格上讨论了半离散和全离散格式下电报方程的类Wilson非协调有限元逼近.利用该元在H1模意义下O(h2)阶的相容误差结果,平均值理论和关于时间t的导数转移技巧得到了超逼近性.进而,借助于插值后处理方法导出了超收敛结果.又由于该元在H1模意义下的相容误差可以达到O(h3)阶,构造了新的外推格式,给出了比传统误差估计高两阶的外推估计.最后,对于给出的全离散逼近格式得到了最优误差估计.     

MAXIMUM NORM ERROR ESTIMATES OF CROUZEIX-RAVIART NONCONFORMING FINITE ELEMENT APPROXIMATION OF NAVIER-STOKES PROBLEM

1. IntroductionThere are many research works on finite element approximation of Navier-Stokesproblem in the case of lower Reynold number, by using the so-called velocity--pressuremixed by Teman [26], the optimal results were also obtained. The other nonconforming finiteelement schemes for Navie--Stokes problem may be found in [4,8,9,14,15,23,26]. But sofar, maximum norm error estimates for any nonconforming finite element schemes werenot considered.Recently, the quasi--optimal maximum norm …     

Analysis on a Finite Volume Element Method for Stokes Problems

Finite volume element method for the Stokes problem is considered. We use a conforming piecewise linear function on a fine grid for velocity and piecewise constant element on a coarse grid for pressure. For general triangulation we prove the equivalence of the finite volume element method and a saddle-point problem, the inf-sup condition and the uniqueness of the approximation solution. We also give the optimal order H^1 norm error estimate. For two widely used dual meshes we give the L^2 norm error estimates, which is optimal in one case and quasi-optimal in another ease. Finally we give a numerical example.     

Stokes方程的压力梯度局部投影间断有限元法

本文对定常的Stokes方程提出了一种新的间断有限元法,通过将通常的间断Galerkin有限元法与压力梯度局部投影相结合,建立了一个稳定的间断有限元格式,对速度和压力的任意分片多项式空间P_l(K),P_m(K)的间断有限元逼近证明了解的存在唯一性,给出了关于速度和压力的L~2范数的最优误差估计.     

Optimal L^2, H^1 and L^\infty analysis of finite volume methods for the stationary Navier–Stokes equations with large data

Previous work on the stability and convergence analysis of numerical methods for the stationary Navier–Stokes equations was carried out under the uniqueness condition of the solution, which required that the data be small enough in certain norms. In this paper an optimal analysis for the finite volume methods is performed for the stationary Navier–Stokes equations, which relaxes the solution uniqueness condition and thus the data requirement. In particular, optimal order error estimates in the $H^1$ -norm for velocity and the $L^2$ -norm for pressure are obtained with large data, and a new residual technique for the stationary Navier–Stokes equations is introduced for the first time to obtain a convergence rate of optimal order in the $L^2$ -norm for the velocity. In addition, after proving a number of additional technical lemmas including weighted $L^2$ -norm estimates for regularized Green’s functions associated with the Stokes problem, optimal error estimates in the $L^\infty $ -norm are derived for the first time for the velocity gradient and pressure without a logarithmic factor $O(|\log h|)$ for the stationary Naiver–Stokes equations.     

One‐level and two‐level finite element methods for the Boussinesq problem

Previous works on the convergence of numerical methods for the Boussinesq problem were conducted, while the optimal L²‐norm error estimates for the velocity and temperature are still lacked. In this paper, the backward Euler scheme is used to discrete the time terms, standard Galerkin finite element method is adopted to approximate the variables. The MINI element is used to approximate the velocity and pressure, the temperature field is simulated by the linear polynomial. Under some restriction on the time step, we firstly present the optimal L² error estimates of approximate solutions. Secondly, two‐level method based on Stokes iteration for the Boussinesq problem is developed and the corresponding convergence results are presented. By this method, the original problem is decoupled into two small linear subproblems. Compared with the standard Galerkin method, the two‐level method not only keeps good accuracy but also saves a lot of computational cost. Finally, some numerical examples are provided to support the established theoretical analysis.     

Navier-Stokes方程对称破坏分歧点的谱Galerkin逼近[英文]

本文研究了Navier-Stokes方程对称破坏分歧点的谱Galerkin逼近问题，构造了定常Navier-Stokes方程对称破坏分歧点扩充系统及其谱Galerkin逼近扩充系统，证明了谱Galerkin逼扩充系统解的存在性和收敛性，从而给出了Navier-Stokes方程对称破坏分歧点的谱Galerkin逼近，并给出了逼近的误差估计。     

On numerical implementations of a new iterative method with boundary condition splitting for solving the nonstationary stokes problem in a strip with periodicity condition

Based on finite-difference approximations in time and a bilinear finite-element approximation in spatial variables, numerical implementations of a new iterative method with boundary condition splitting are constructed for solving the Dirichlet initial-boundary value problem for the nonstationary Stokes system. The problem is considered in a strip with a periodicity condition along it. At each iteration step of the method, the original problem splits into two much simpler boundary value problems that can be stably numerically approximated. As a result, this approach can be used to construct new effective and stable numerical methods for solving the nonstationary Stokes problem. The velocity and pressure are approximated by identical bilinear finite elements, and there is no need to satisfy the well-known difficult-to-verify Ladyzhenskaya-Brezzi-Babuska condition, as is usually required when the problem is discretized as a whole. Numerical iterative methods are constructed that are first- and second-order accurate in time and second-order accurate in space in the max norm for both velocity and pressure. The numerical methods have fairly high convergence rates corresponding to those of the original iterative method at the differential level (the error decreases approximately 7 times per iteration step). Numerical results are presented that illustrate the capabilities of the methods developed.     

An Error Analysis of Discontinuous Finite Element Methods for the Optimal Control Problems Governed by Stokes Equation

In this article, an abstract framework for the error analysis of discontinuous finite element method is developed for the distributed and Neumann boundary control problems governed by the stationary Stokes equation with control constraints. A priori error estimates of optimal order are derived for velocity and pressure in the energy norm and the L²-norm, respectively. Moreover, a reliable and efficient a posteriori error estimator is derived. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. In particular, we consider the abstract results with suitable stable pairs of velocity and pressure spaces like as the lowest-order Crouzeix–Raviart finite element and piecewise constant spaces, piecewise linear and constant finite element spaces. The theoretical results are illustrated by the numerical experiments.     

On Two Iterative Least-Squares Finite Element Schemes for the Incompressible Navier–Stokes Problem

This paper is mainly devoted to a comparative study of two iterative least-squares finite element schemes for solving the stationary incompressible Navier–Stokes equations with velocity boundary condition. Introducing vorticity as an additional unknown variable, we recast the Navier–Stokes problem into a first-order quasilinear velocity–vorticity–pressure system. Two Picard-type iterative least-squares finite element schemes are proposed to approximate the solution to the nonlinear first-order problem. In each iteration, we adopt the usual L ² least-squares scheme or a weighted L ² least-squares scheme to solve the corresponding Oseen problem and provide error estimates. We concentrate on two-dimensional model problems using continuous piecewise polynomial finite elements on uniform meshes for both iterative least-squares schemes. Numerical evidences show that the iterative L ² least-squares scheme is somewhat suitable for low Reynolds number flow problems, whereas for flows with relatively higher Reynolds numbers the iterative weighted L ² least-squares scheme seems to be better than the iterative L ² least-squares scheme. Numerical simulations of the two-dimensional driven cavity flow are presented to demonstrate the effectiveness of the iterative least-squares finite element approach.     

On the solution of a Schrödinger type equation by a projection-difference method with an implicit Euler scheme with respect to time

A linear nonstationary Schrödinger type problem in a separable Hilbert space is approximately solved by a projection-difference method. The problem is discretized in space by the Galerkin method using finite-dimensional subspaces of finite-element type, and an implicit Euler scheme is used with respect to time. We establish error estimates uniform with respect to the time grid for the approximate solutions; as to the spatial variables, the estimates are given in the norm of the original space as well as in the energy norm. The estimates considered here not only permit one to prove the convergence of approximate solutions to the exact solution but also give a numerical characterization of the convergence rate.