高次三角形有限元外推的探讨

探讨泊松方程高次三角形有限元外推公式.为此先推导离散格林函数的权模估计和有限元解的渐近不等式展开,然后给出公式的证明.     

四阶方程两点边值问题Hermite有限元解的渐近展式与外推

1引言有限元解的渐近展式是提高微分方程数值解精度的重要工具,比如亏量校正和外推就是建立在有限元解的渐近展式的基础之上．许多作者对此进行了大量的研究(见[1]-[4]),特别是文[1],提出了在研究有限元解的渐近展式中十分有用的能量嵌入技巧．本文利用能量嵌入定理得到了四阶方程两点边值问题Hermite有限元解及其二阶平均导数的渐近展式,进一步我们还讨论了它们的Richardson外推公式．考虑四阶方程两点边值问题     

n维矩形域上椭圆问题有限元单方向外推

1 引言 Richardson外推应用于椭圆偏微方程边值问题有限元法始于1978年(见[1],并于1983年在理论研究方面取得突破性进展(见[2]).自那以后有限元外推得到迅速发展,成为一个富于竞争的国际性研究课题(见[3],[4],[5]及其所列参考文献).但是通常的有限元外推需同时在每一个方向上分半加密网格,因此,对n维问题,细网格的结点数是粗网格的2~n倍,结果当n较大时(高维问题),细网格上的计算工作量十分庞大.为了克服这个缺点,发展了有限元单方向外推.对Poisson方程边值问题,[6]研究了2维矩形域上双线性有限元解的单方向外推,[7]研究了3维矩形域上三线性有限元单方向外推必须的插值渐近展开式,[8]研究了n维矩形域上n线性有限元解的区域分裂外推.本文旨在研究n维矩形域上Poisson方程边值问题及其对应的本征值问题n线性有限元解的单方向外推.始终假设本文出现的函数u是连续的.     

非饱和土壤水流方程基于特征投影分解方法的降阶CN有限元外推模型

利用Crank-Nicolson有限元方法和特征投影分解方法去建立二维非饱和土壤水流方程的一种维数很低,精度足够高的降阶CN有限元外推模型,并给出这种降阶CN有限元外推模型的降阶近似解误差估计和算法实现.最后用数值例子说明数值结果与理论结果相吻合,并阐明这种降阶CN有限元外推模型的优越性.     

拟线性Sobolev方程Carey元解的高精度分析

在一种半离散格式下讨论了拟线性Sobolev方程Carey元的超收敛及外推.根据Carey元的构造证明了其有限元解的线性插值与三角形线性元的解相同,再结合线性元的高精度分析和插值后处理技巧导出了超逼近和整体超收敛及后验误差估计.与此同时,根据线性元的误差渐近展开式,构造了一个新的辅助问题,得到了比传统的有限元误差高三阶的外推结果.     

二维土壤溶质输运方程基于POD方法的降阶CN有限元外推算法

首先给出二维土壤溶质输运方程时间二阶精度的Crank-Nicolson(CN)时间半离散化格式和时间二阶精度的全离散化CN有限元格式及其误差分析.然后利用特征投影分解(proper orthogonal decomposition,简记为POD)方法对二维土壤溶质输运方程的经典CN有限元格式做降阶处理,建立一种具有足够高精度、自由度很少的降阶CN有限元外推格式,并给出这种降阶CN有限元解的误差估计和外推算法的实现.最后用数值例子说明数值结果与理论结果是相吻合的.     

Stokes问题Q_2-P_1混合元外推方法

考虑Stokes问题的有限元解与精确解插值的Q2-P1混合元的渐近误差展开和分裂外推.首先利用积分恒等式技巧确定了微分方程精确解与有限元插值之间积分式的主项,其次再借助插值后处理和分裂外推技术,得到了比通常的误差估计提高两阶的收敛速度.     

高次有限元外推的进一步研究

对高次矩形元,我们给出了一个外推算法.利用离散格林函数权模估计和渐进不等式展开,证明了外推算法可以提高泊松方程有限元逼近解的精度.     

非对称不定问题类Wilson元的超收敛和外推

讨论了非对称不定问题的类Wilson有限元逼近.利用该元的特殊性并借助于双线性元已有的高精度分析结果和平均值技巧,得到了O(h~2)阶的超逼近和整体超收敛结果,同时给出了新的渐进展开式,导出了O(h~3)阶的外推解,这比传统的误差估计高两阶.     

本征值问题有限元近似解的外推方法

§1.引言利用外推算法提高有限元近似解的精度,我们曾对二阶椭圆型问题进行研究,本文中将继[1]之后,讨论本征值问题有限元解的外推方法.不失一般性,仅讨论下述模型问题,但本文方法和结论可以推广到一般二阶椭圆型偏微分算子上.考虑     

Analysis of extrapolation cascadic multigrid method(EXCMG)

Based on an asymptotic expansion of finite element,a new extrapolation formula and extrapolation cascadic multigrid method(EXCMG)are proposed,in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid.In the case of triple grids,the error of the new initial value is analyzed in detail.A larger scale computation is completed in PC.     

L^2-ERROR OF EXTRAPOLATION CASCADIC MULTIGRID （EXCMG）

Based on an asymptotic expansion of finite element, an extrapolation cascadic multigrid method （EXCMG） is proposed, in which the new extrapolation and quadratic interpolation are used to provide a better initial value on refined grid. In the case of multiple grids, both superconvergence error in H^1-norm and the optimal error in l2-norm are analyzed. The numerical experiment shows the advantage of EXCMG in comparison with CMG.     

An algorithm using the finite volume element method and its splitting extrapolation

This paper is to present a new efficient algorithm by using the finite volume element method and its splitting extrapolation. This method combines the local conservation property of the finite volume element method and the advantages of splitting extrapolation, such as a high order of accuracy, a high degree of parallelism, less computational complexity and more flexibility than a Richardson extrapolation. Because the splitting extrapolation formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid, a large scale multidimensional problem is turned into a set of smaller discrete subproblems. Additionally, this method is efficient for solving interface problems if we regard the interfaces of the problems as the interfaces of the initial domain decomposition.     

Accuracy Analysis of the Adini Element for Biharmonic Equation

In this paper,we consider the solution of the biharmonic equation using Adini nonconforming finite element,and report new results of the asympiotic error expansions of the interpolation error functionals and nonconforming remainder.These expansions are used to develop two extrapolation formulas and a series of sharp error estimates.Finally,the numerical results have verified the extrapolation theory.     

不可压缩流动的并行数值方法

不可压缩流动的数值模拟是计算流体力学的重要组成部分. 基于有限元离散方法, 本文设计了不可压缩Navier-Stokes (N-S)方程支配流的若干并行数值算法. 这些并行算法可归为两大类: 一类是基于两重网格离散方法, 首先在粗网格上求解非线性的N-S方程, 然后在细网格的子区域上并行求解线性化的残差方程, 以校正粗网格的解; 另一类是基于新型完全重叠型区域分解技巧, 每台处理器用一局部加密的全局多尺度网格计算所负责子区域的局部有限元解. 这些并行算法实现简单, 通信需求少, 具有良好的并行性能, 能获得与标准有限元方法相同收敛阶的有限元解. 理论分析和数值试验验证了并行算法的高效性     

Extrapolation of Mixed Finite Element Approximations for the Maxwell Eigenvalue Problem

In this paper,a general method to derive asymptotic error expansion formulas for the mixed finite element approximations of the Maxwell eigenvalue problem is established.Abstract lemmas for the error of the eigenvalue approximations are obtained.Based on the asymptotic error expansion formulas,the Richardson extrapolation method is employed to improve the accuracy of the approximations for the eigenvalues of the Maxwell system from θ(h2) to θ(h4) when applying the lowest order Nédé1ec mixed finite element and a nonconforming mixed finite element.To our best knowledge,this is the first superconvergence result of the Maxwell eigenvalue problem by the extrapolation of the mixed finite element approximation.Numerical experiments are provided to demonstrate the theoretical results.     

A high‐order finite difference discretization strategy based on extrapolation for convection diffusion equations

We propose a new high‐order finite difference discretization strategy, which is based on the Richardson extrapolation technique and an operator interpolation scheme, to solve convection diffusion equations. For a particular implementation, we solve a fine grid equation and a coarse grid equation by using a fourth‐order compact difference scheme. Then we combine the two approximate solutions and use the Richardson extrapolation to compute a sixth‐order accuracy coarse grid solution. A sixth‐order accuracy fine grid solution is obtained by interpolating the sixth‐order coarse grid solution using an operator interpolation scheme. Numerical results are presented to demonstrate the accuracy and efficacy of the proposed finite difference discretization strategy, compared to the sixth‐order combined compact difference (CCD) scheme, and the standard fourth‐order compact difference (FOC) scheme. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 18–32, 2004.     

三维泊松方程基于Richardson外推法的高阶紧致差分方法

基于Richardson外推法提出了数值求解三维泊松方程的高阶紧致差分方法.方法通过利用四阶和六阶紧致差分格式,分别在细网格和粗网格上求解,然后利用Richardson外推技术和算子插值方法,得到三维泊松方程在细网格上的六阶和八阶精度的数值解.数值实验结果验证了该方法的精确性和有效性.     

Cascadic multigrid methods combined with sixth order compact scheme for poisson equation

Based on the extrapolation theory and a sixth order compact difference scheme, new extrapolation interpolation operator and extrapolation cascadic multigrid methods for two dimensional Poisson equation are presented. The new extrapolation interpolation operator is used to provide a better initial value on refined grid. The convergence of the new methods are given. Numerical experiments are shown to illustrate that the new methods have higher accuracy and efficiency.