线性对流占优扩散问题的交替方向差分流线扩散法

本文将交替方向法与差分流线扩散法(简称FDSD方法)相结合,对于二维线性对流占优扩散问题构造了一种交替方向差分流线扩散格式,给出了格式的实现过程并就稳定性及误差进行了分析．此格式不但实现了对数值求解二维对流扩散方程降维的目的,并且保持了FDSD方法良好的稳定性及高精度阶的基本性质．最后给出数值算例说明算法的有效性．     

非线性对流扩散问题的交替方向差分流线扩散法

1引言Peaceman,Douglas等人于1955年提出了差分格式的交替方向法。随后,Douglas,Dupont于1972年又提出了有限元格式的交替方向法[1]。其基本思想是:对两个或三个空间变量的二阶抛物型和双曲型问题,将交替方向法与Galerkin方法相结合,通过算子分裂技术,把高维问题转化为一系列低维问题,交替地沿各空间变量的方向求解。[2]、[3]和[4]给出了对更一般扩散问题(带对流项的抛物方程)的数值求解和误差分析。     

一类非定常对流占优扩散问题差分-流线扩散法的后处理

讨论了一类非定常对流占优扩散方程的差分-流线扩散格式(FDSD),利用插值后处理技术,提高了特殊网格下该FDSD格式在双线性元空间的精度,从而按L∞(L2(Ω) 模达到最优．     

THE FINITE DIFFERENCE STREAMLINE DIFFUSION METHODS FOR TIME-DEPENDENT CONVECTION-DIFFUSION EQUATIONS

In this paper, two finite difference streamline diffusion (FDSD) schemes for solving two-dimensional time-dependent convection-diffusion equations are constructed. Stability and optimal order error estimati-ions for considered schemes are derived in the norm stronger than L~2-norm.     

非定常对流扩散问题差分-流线扩散法的线性三角元解的最优误差估计

讨论了差分-流线扩散法(FDSD)求解线性对流占优扩散问题解的精度,利用插值后处理技术,使该格式解的空间精间达到最优.     

对流扩散问题的交替方向差分-流线扩散格式

1.引 言 差分-流线扩散法（Finite Difference-Streamline Diffusion Method,简称FDSD方法）于1998年由文[1]提出并对线性对流占优扩散问题给出分析,随后文[2],[3]就非线性问题的FDSD格式及FDSD预测-校正格式,分别作出了分析,文[4]讨论了FDSD方法的后验估计及自适应技术,[5],[6]则分别讨论了FDSD方法的某些重要应用.与基于时-空有限元的传统流线扩散法相比,FDSD方法的计算工作量已有成数量级的减少,且较易于推广到非线性问题,然而,对于高维问题,在每一时间层,仍然需要求解一大型线性或非线性方程组,工作量仍然很大.参照J.Douglas与T.Dupont关于抛物问题交替方向     

Streamline diffusion finite element method for stationary incompressible magnetohydrodynamics

In this article, a streamline diffusion finite element method is proposed and analyzed for stationary incompressible magnetohydrodynamics (MHD) equations. This method is stable for any combinations of velocity, pressure, and magnet finite element spaces, without requiring Ladyzenskaja‐Babu?ka‐Brezzi (LBB) condition. The well‐posedness and convergence (at optimal error rate) of this scheme are proved in terms of some conditions. Two numerical experiments are illustrated to validate our theoretical analysis and show the streamline diffusion finite element approach is effective for solving the MHD problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1877–1901, 2014     

对流占优扩散问题的经济型流线扩散有限元法

In this paper, the economical finite difference-streamline diffusion (EFDSD) schemes based on the linear F.E. space for time-dependent linear and non-linear convection-dominated diffusion problems are constructed. The stability and error estimation with quasi-optimal order approximation are established in the norm stronger than L^2 - norm for the schemes considered. It is indicated by the results obtained that,for linear F.E. space, the EFDSD schemes have the same specific properties of stability and convergence as the traditional FDSD schemes for the problems discussed.     

Pointwise error estimates for a streamline diffusion finite element scheme

Summary Pointwise error estimates for a streamline diffusion scheme for solving a model convection-dominated singularly perturbed convection-diffusion problem are given. These estimates improve pointwise error estimates obtained by Johnson et al.[5].     

一个新非协调单元对扩散对流反应方程的应用

利用最近提出的一个新型非协调双参数单元,将流线扩散有限元方法成功地应用于对流占优的扩散对流反应方程,并且得到流线扩散模意义下的误差估计结果.     

对流扩散问题Crank-Nicolson差分-流线扩散法的高精度分析

讨论了对流扩散问题C rank-N ico lson差分流线扩散格式,利用插值后处理技术提高了特殊网格下该格式在双线性元空间解的精度,从而按Lα(L2(Ω))模达到最优.     

二维对流扩散方程满足最大值原理的MMOCAA差分方法

Using FCT idea,the non-oscillation MMOCAA(The modified method of characteristics with adjusted advection) finite difference scheme satisfing the discrete maximum principle for convection-dominated diffusion equation in 2D is constructed.The scheme is free from oscillation,with which the problem is solved by the MMOCAA difference method based on 2-order Lag-range interpolation proposed by Jim.Douglas, Jr.(Numer.Math.,1999,83:353-369.). The error analysis of the new scheme and numerical example are given in the paper.The numerical example shows that the scheme has smaller numerical viscosity than the MMOCAA difference method based on bilineax Lagrange interpolation.     

Three-step UNO-MMOCAA Difference Method for Convection Diffusion Equation

By combing the three-step modified method of characteristics and MMO-CAA difference method with UNO interpolation, the three-step UNO-MMOCAA finite difference method is established for convection-dominated diffusion problems in this paper. The scheme is two-order accurate in space and time and is free from the     

A finite difference analysis of a streamline diffusion method on a Shishkin mesh

We consider streamline diffusion finite element methods applied to a singularly perturbed convection–diffusion two‐point boundary value problem whose solution has a single boundary layer. To analyse the convergence of these methods, we rewrite them as finite difference schemes. We first consider arbitrary meshes, then, in analysing the scheme on a Shishkin mesh, we consider two formulations on the fine part of the mesh: the usual streamline diffusion upwinding and the standard Galerkin method. The error estimates are given in the discrete L _∞ norm; in particular we give the first analysis that shows precisely how the error depends on the user-chosen parameter _τ0 specifying the mesh. When _τ0 is too small, the error becomes O(1), but for _τ0 above a certain threshold value, the error is small and increases either linearly or quadratically as a function of . Numerical tests support our theoretical results. This revised version was published online in August 2006 with corrections to the Cover Date.     

对流扩散问题的Crank-Nicolson差分-流线扩散法

1　引　言Streamline- Diffusion method (SD方法 )是近年来 Hughes和 Brooks提出的一种求解定常的对流占优和对流扩散问题的人工粘性有限元方法[1 ] ,[2 ] ,它具有标准有限元方法的高阶精度特点和人工粘性 Galerkin方法的稳定性特点 ,因此越来越受到人们的重视 .现在 ,SD方法已被推广到 Euler方程和 Navier- Stokes方程等发展型对流扩散问题[3 ] [4] ,但是常常采用时空有限元 [3 ] [5] ,这样能把时间和空间的精度很好地统一起来 ,却增大了数值计算的复杂性 ,基于此 [6 ]对非线性的对流占优扩散问题提出一种 Finite Difference- Strea…     

Multilevel a posteriori error analysis for reaction–convection–diffusion problems

We present a new approach to the a posteriori error analysis of stable Galerkin approximations of reaction–convection–diffusion problems. It relies upon a non-standard variational formulation of the exact problem, based on the anisotropic wavelet decomposition of the equation residual into convection-dominated scales and diffusion-dominated scales. The associated norm, which is stronger than the standard energy norm, provides a robust (i.e., uniform in the convection limit) control over the streamline derivative of the solution. We propose an upper estimator and a lower estimator of the error, in this norm, between the exact solution and any finite dimensional approximation of it. We investigate the behaviour of such estimators, both theoretically and through numerical experiments. As an output of our analysis, we find that the lower estimator is quantitatively accurate and robust.     

一维定常型对流占优扩散方程的一类迎风有限体积格式

本文针对一维定常型对流占优扩散方程提出了一类迎风有限体积格式 .该格式对对流项具有二阶精度 ,对扩散项保持一阶精度 ,符合对流占优扩散问题强对流、弱扩散的特点 .     

A stabilized finite element method for convection–diffusion problems

A stabilized finite element method (FEM) is presented for solving the convection–diffusion equation. We enrich the linear finite element space with local functions chosen according to the guidelines of the residual‐free bubble (RFB) FEM. In our approach, the bubble part of the solution (the microscales) is approximated via an adequate choice of discontinuous bubbles allowing static condensation. This leads to a streamline‐diffusion FEM with an explicit formula for the stability parameter τ_K that incorporates the flow direction, has the capability to deal with problems where there is substantial variation of the Péclet number, and gives the same limit as the RFB method. The method produces the same a priori error estimates that are typically obtained with streamline‐upwind Petrov/Galerkin and RFB. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

三维热传导型半导体器件瞬态模拟问题的Crank-Nicolson差分-流线扩散有限元法及其数值分析

本文研究三维热传导型半导体器件瞬态模拟问题的数值方法.针对数学模型中各方程不同的特点,分别提出不同的有限元格式.特别针对浓度方程组是对流为主扩散问题的特点,使用Crank-Nicolson差分-流线扩散计算格式,提高了数值解的稳定性.得到的L2误差估计关于空间剖分步长是拟最优的,关于时间步长具有二阶精度.     

对流扩散方程的稳定化流线扩散法最小二乘非协调有限元分析

将最小二乘法和稳定化的流线扩散法相结合,研究了对流扩散方程的非协调有限元格式,用矩形EQ_1~(rot)元和零阶R-T元分别来逼近位移和应力,利用单元本身的特殊性质,证明了离散格式解的存在惟一性,得到了位移H~1-模和应力H(div)-模的最优误差估计.