抛物方程的一种广义差分法(有限体积法)

广义差分法自1982年被提出,至今已获得很大发展（见[1]或[10],这种方法在国际上被称为有限体积（元）法（见[8],[9]）,它的主要优点是保持物理量的局部守恒性.文[3],[5]分别将三角形网格上的椭圆型方程的广义差分法（有限体积法）（见[2],[4]）推广到抛物型方程.我们知道三角形网格与四边形网格是两种基本的分割空间区域的方法,实践上使用哪一种网格,要根据空间区域的几何形状而定.文[7],[6]讨论了一般四边形网上椭圆型方程的广义差分法.本文以抛物方程为模型,取试探函数空间为一般四边形剖分上的等参双线性元,检验函数空间为对偶剖分上的分片常数,导出了一种新的有效的广义差分算法（有限体积算法）,证明了半离散与全离散格式的最佳H1误差估计.遇到的主要困难是双线性形式a（uh,Πh＊uh）     

一类与BB型对偶剖分相关的广义差分格式的L2模误差估计

广义差分法的研究涉及到对求解区域做某种剖分及相应的对偶剖分,对区域做三角形剖分时,最常用的对偶剖分是所谓的外心对偶剖分和重心对偶剖分[4][2],在计算流体中,还经常用到另一种对偶剖分即重心相联对偶剖分(BB型对偶剖分)[6]。[6]对一类椭圆     

一类与BB型对偶剖分相关的广义差分格式

豆引言考虑二阶椭圆方程的边值问题〔,a.an.a.＆t-。^D一[六（a于）十十…于）」一八x,y）,（x,y）En,（1.1）J“山一而“av”-av”。。、一、。。,\一·。·---,l。。、____（.2）LU一0,（J,y）E厂一go,其中D是多边形区域,厂一JD为0的边界,f（x,y）为o上的已知函数/三L‘（0）,系数a—a（xJ）充分光滑,且存在常数7>O,使得以X,y）>/,对任意（X,y）ED.文[1],[2]已系统地研究了求解口.1）,（l.2）的广义差分法.广义差分法的基本思想是对求解区域0作三角形剖分TA及其对偶剖分T;,然后构造T^上的试探函数空间U。（有限元空间）和耳上的…     

广义平稳高斯场的可离散化性

§1 引言和定义近几年来,人们对离散参数的随机场的所谓自相似性进行了研究,在这方面,有许多有趣的结果(见[1],[5],[6])。Dobrushin在[2],[3]中引入了广义自相似随机场的概念,并对它进行了详细的研究。从几方面讲,这种定义似乎更自然,更有用。首先,在一定条件下,离散自相似场可以被嵌入一个广义自相似场;其次,可以通过广义自相似场的离散化来构造离散自     

关于非线性双曲波动变分方程的动力学形式

受到[3],[4]和[5]的启发,本文对应于某种波动变分方程的弱解,给出了该方程的动力学 形式,此方程来源于长原子液晶运动双极链中的长波以及其它邻域的研究.     

加罚N-S方程的有限元非线性Galerkin方法

非线性Galerkin方法是对耗散型非线性发展方程的一种数值解法,其空间变量不象一般Galerkin方法那样在线性空间上离散,而是在非线性流形上离散,所得逼近解在时间变量增大时可以更快地逼近其精确解.精细的理论分析可见[1],[2]等,在有限元逼近基础上将此方法应用到Navier-Stokes方程上的工作可参见[3],[4],这些文章主要针对速度与压力同时求解的混合元情形做了讨论.本文在[4]的基础上对加罚Navier-Stokes方程的一种非线性Galerkin方法的半离散和全离散有限元逼近格式分别进行了误差估     

加罚N-S方程的有限元非线性Galerkin方法

非线性Galerkin方法是对耗散型非线性发展方程的一种数值解法,其空间变量不象一般Galerkin方法那样在线性空间上离散,而是在非线性流形上离散,所得逼近解在时间变量增大时可以更快地逼近其精确解.精细的理论分析可见[1],[2]等,在有限元逼近基础上将此方法应用到Navier-Stokes方程上的工作可参见[3],[4],这些文章主要针对速度与压力同时求解的混合元情形做了讨论.本文在[4]的基础上对加罚Navier-Stokes方程的一种非线性Galerkin方法的半离散和全离散有限元逼近格式分别进行了误差估     

抛物方程的基于BB型对偶剖分的有限体积元法

本文讨论了抛物方程的基于三角形剖分和BB型对偶剖分的有限体积元法,给出了半离散及全离散有限体积元格式的最佳阶L2和H1误差估计.     

非线性广义系统最优控制的最大值原理：无限维情形

§1.引言 对于无限维非线性最优控制问题,[1]—[3]在一定条件下证明了最大值原理。在有限维情形,[4]讨论了线性广义系统的二次型指标最优问题。关于有限维非线性广义系统的讨论见[5],[6]。而对于无限维非线性广义系统的最优控制问题,目前尚无讨论。本文利用Ekeland变分原理[7]—[10]和Fattorini引理,对具有一般目标泛函的无限维广义系统的最优控制问题给出了最大值原理。     

一个退化的变分不等式的数值解

1.引言 稳态渗流自由边界问题大部可化为椭圆型变分不等式或拟变分不等式,其数值解已为不少作者所研究(参看[1],[2]及其文献)。某些轴对称问题则可化为另一类变分不等式一退化的椭圆型变分不等式。[3],[4]和[5]即用此法研究轴对称渗流井(水     

拟线性拋物方程半离散变网格有限元方法的误差估计

§1.导言近年来,变网格方法正日益为人们所重视与应用,但理论性分析文献仍不多见。文献[1]讨论了某些发展型方程变网格方法的误差估计,但未给出收敛阶估计;文献[2,3]仅对全离散方法讨论了收敛阶问题。本文对一类拟线性抛物问题,于第二节中给出了半离散Galerkin变网格计算格式及其可解性定理;第三节中建立了对称误差估计;第四节给     

A posteriori error analysis for numerical approximations of Friedrichs systems

The global error of numerical approximations for symmetric positive systems in the sense of Friedrichs is decomposed into a locally created part and a propagating component. Residual-based two-sided local a posteriori error bounds are derived for the locally created part of the global error. These suggest taking the -norm as well as weaker, dual norms of the computable residual as local error indicators. The dual graph norm of the residual is further bounded from above and below in terms of the norm of where h is the local mesh size. The theoretical results are illustrated by a series of numerical experiments. Received January 10, 1997 / Revised version received March 5, 1998     

非线性抛物型方程的变网格有限元法

Three linear finite element schemes with moving mesh are established for a class of nonlinear parabolic equations.Optimal L^2-norm and energy norm error estimates are derived under certain conditions.     

Analysis of finite element methods for second order boundary value problems using mesh dependent norms

This paper presents a new approach to the analysis of finite element methods based onC ⁰-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to theL ₂ norm and to the 2nd order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis in which error estimates are first derived in the 1st order Sobolev norm and subsequently are derived in theL ₂ norm and in the 2nd order Sobolev norm — the 2nd order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the 2nd order Sobolev space and that the mesh family is quasi-uniform.     

不可压缩核废料污染问题的变网格特征有限元法

1 引言 多孔介质中的核废料污染问题是环境保护领域的重要课题。对于不可压缩二维模型,它是地层中迁移型耦合抛物型方程组的初边值问题:     

NUMERICAL METHODS FOR MISCIBLE DISPLACEMENT IN POROUS MEDIA INFLUENCED BY MOBILE WATER AND IMMOBILE WATER

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Finite-Difference Methods for a Class of Strongly Nonlinear Singular Perturbation Problems

The paper is concerned with strongly nonlinear singularly perturbed bound- ary value problems in one dimension.The problems are solved numerically by finite- difference schemes on special meshes which are dense in the boundary layers.The Bakhvalov mesh and a special piecewise equidistant mesh are analyzed.For the central scheme,error estimates are derived in a discrete L~1 norm.They are of second order and decrease together with the perturbation parameterε.The fourth-order Numerov scheme and the Shishkin mesh are also tested numerically.Numerical results showε-uniform pointwise convergence on the Bakhvalov and Shishkin meshes.     

Analysis and convergence of finite volume method using discontinuous bilinear functions

We develop finite volume method using discontinuous bilinear functions on rectangular mesh. This method is analyzed for the Stokes equations. An optimal error estimate for the approximation of velocity is obtained in a mesh‐dependent norm. First order L²‐error estimates are derived for the approximations of both velocity and pressure. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

非线性双曲型方程的变网格有限元法

对一类非线性双曲型方程给出了两种变网格有限元逼近格式 .在一定条件下 ,得到了最优 H 1模误差估计     

An iterative method with error estimators

Iterative methods for the solution of linear systems of equations produce a sequence of approximate solutions. In many applications it is desirable to be able to compute estimates of the norm of the error in the approximate solutions generated and terminate the iterations when the estimates are sufficiently small. This paper presents a new iterative method based on the Lanczos process for the solution of linear systems of equations with a symmetric matrix. The method is designed to allow the computation of estimates of the Euclidean norm of the error in the computed approximate solutions. These estimates are determined by evaluating certain Gauss, anti-Gauss, or Gauss–Radau quadrature rules.