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.     

Sobolev型方程Wilson元解的高精度分析

本文利用积分恒等式和插值后处理等技术对 Sobolev型方程 Wilson非协调有限元解进行了高精度算法分析 ,获得了解的超逼近性质和插值有限元解的整体超收敛 .在此基础上 ,运用外推与校正方法进一步获得了具有更高精度的近似解及后验误差估计 .     

基于区域分解的双调和方程的混合元近似解的分裂外推

本文主要讨论了基于区域分解的双调和方程的混合元近似解的分裂外推。并得到解的多变量展开。     

奇异摄动问题的一致收敛离散方法解的一致高阶精度外推

本文讨论基于整体误差一致展开式的一致收敛离散方法解的一致高阶精度外推.将该方法应用于非自共轭问题的Il'in-Allen-Southwell格式,我们得到了二阶一致收敛的外推解,并用数值计算说明该结论.     

带有小波函数积分的外推加速算法

数值计算光滑函数与小波函数的内积,是小波在数值分析的应用中经常遇到的一个典型问题,本文给出了一种外推积分方法,可以行之有效地提高数值解精度,减少计算量     

交通流模型基于特征投影分解技术的外推降维有限差分格式

利用特征正交分解(proper orthogonal decomposition,简记为POD)技术研究交通流的Aw-Rascle-Zhang(ARZ)模型. 建立一种基于 POD方法维数较低的外推降维有限差分格式, 并用数值例子检验数值计算结果与理论结果相吻合, 进一步表明基于POD方法的外推降维有限差分格式对于求解交通流方程数值解是可行和有效的.     

平面双调和问题的第一类边界积分方程的高精度求积方法与外推

借助位势理论,平面双调和方程的Dirichlet问题被转化为第一类边界积分方程组,本文使用新型的反常积分的求积公式构造出解造解此类边界积分方程的机械求积方法,证明了该方法具有O（h^3)阶精度和误差的h^3幂渐近展开,故借助Richardson外推还能提高精度阶。     

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.     

三维热传导方程的紧交替方向差分格式

本文研究了三维热传导方程的紧交替方向隐式差分格式.利用算子方法导出了紧交替方向隐式差分格式,并利用Fourier分析方法证明了差分格式的收敛性和绝对稳定性,Richardson外推法外推一次得到具有O(T3+h6)阶精度的近似解.本文方法是对二维热传导方程问题的推广,同样适用于多维的情形.     

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.     

An extrapolation method for a class of boundary integral equations

Boundary value problems of the third kind are converted into boundary integral equations of the second kind with periodic logarithmic kernels by using Green's formulas. For solving the induced boundary integral equations, a Nyström scheme and its extrapolation method are derived for periodic Fredholm integral equations of the second kind with logarithmic singularity. Asymptotic expansions for the approximate solutions obtained by the Nyström scheme are developed to analyze the extrapolation method. Some computational aspects of the methods are considered, and two numerical examples are given to illustrate the acceleration of convergence.

     

Fast Iterative Methods for Solving of Boundary Nonlinear Integral Equations with Singularity

A very efficient and fully discrete method for numerical solution of boundary nonlinear integral equation is described. There seems a lack of rigorous numerical analysis because of singular or hypersingular behavior. In this paper, we suggest variants of methods for solving numerical solutions. Moreover, our aim has been to show how the iterations can be effectively and efficiently regularized for solving ill-posed problems by using the preconditioner. We have compared these methods with CPU time and iterations. Finally, some numerical examples show the efficiency of the proposed methods.     

Difference Potentials Method and its Applications

The purpose of this article is to acquaint the reader with the general concepts and capabilities of the Difference Potentials Method (DPM). DPM is used for the numerical solution of boundary-value and some other problems in difference and differential formulations. Difference potentials and DPM play the same role in the theory of solutions of linear systems of difference equations on multi-dimensional non-regular meshes as the classical Cauchy integral and the method of singular integral equations do in the theory of analytical functions (solutions Cauchy-Riemann system). The application of DPM to the solution of problems in difference formulation forms the first aspect of the method. The second aspect of the DPM implementation is the discretization and numerical solution of the Calderon-Seeley boundary pseudo-differential equations. The latter are equivalent to elliptical differential equations with variable coefficients in the domain; they are written making no use of fundamental solutions and integrals. Because of this fact ordinary methods for discretization of integral equations cannot be applied in this case. Calderon-Seeley equations have probably not been used for computations before the theory of DPM appeared. This second aspect for the implementation of DPM is that it does not require difference approximation on the boundary conditions of the original problem. The latter circumstance is just the main advantage of the second aspect in comparison with the first one. To begin with, we put forward and justify the main constructions and applications of DPM for problems connected with the Laplace equation. Further, we also outline the general theory and applications: both those already realized and anticipated.     

Numerical solution of one-dimensional Burgers’ equation using reproducing kernel function

This paper presents a numerical method for one-dimensional Burgers’ equation by the Hopf–Cole transformation and a reproducing kernel function, abbreviated as RKF. The numerical solution is given as explicit integral expressions with the RKF at each time step, so that the computation is fully parallel. The stability and error estimates are derived. Numerical results for some test problems are presented and compared with the exact solutions. Some numerical results are also compared with the results obtained by other methods. The present method is easily implemented and effective.     

Reduced-order extrapolation spectral-finite difference scheme based on POD method and error estimation for three-dimensional parabolic equation

In this study, a classical spectral-finite difference scheme (SFDS) for the three-dimensional (3D) parabolic equation is reduced by using proper orthogonal decomposition (POD) and singular value decomposition (SVD). First, the 3D parabolic equation is discretized in spatial variables by using spectral collocation method and the discrete scheme is transformed into matrix formulation by tensor product. Second, the classical SFDS is obtained by difference discretization in time-direction. The ensemble of data are comprised with the first few transient solutions of the classical SFDS for the 3D parabolic equation and the POD bases of ensemble of data are generated by using POD technique and SVD. The unknown quantities of the classical SFDS are replaced with the linear combination of POD bases and a reducedorder extrapolation SFDS with lower dimensions and sufficiently high accuracy for the 3D parabolic equation is established. Third, the error estimates between the classical SFDS solutions and the reduced-order extrapolation SFDS solutions and the implementation for solving the reduced-order extrapolation SFDS are provided. Finally, a numerical example shows that the errors of numerical computations are consistent with the theoretical results. Moreover, it is shown that the reduced-order extrapolation SFDS is effective and feasible to find the numerical solutions for the 3D parabolic equation.     

Numerical Verification of Solutions of Periodic Integral Equations with a Singular Kernel

This paper proposes the method of numerical verification of solutions of a periodic integral equation with a logarithmic singular kernel, which is typically found in some two-dimensional potential problems. The verification method utilizes a property of the singular integral for trigonometric polynomials, the periodic Sobolev space and Schauder's fixed point theorem.     

无穷凹角区域各向异性问题的自然边界元法

本文研究无穷凹角区域上一类各向异性问题的自然边界元法.利用自然边界归化原理,获得该问题的Poisson积分公式和自然积分方程,给出了自然积分方程的数值方法,以及逼近解的收敛性和误差估计,最后给出了数值例子,以示方法的可行性和有效性.     

On the combination of some semi‐discretization methods and boundary integral equations for the numerical solution of initial boundary value problems

We consider initial boundary value problems for the homogeneous differential equation of hyperbolic or parabolic type in the unbounded two‐ or three‐dimensional spatial domain with the homogeneous initial conditions and with Dirichlet or Neumann boundary condition. The numerical solution is realized in two steps. At first using the Laguerre transformation or Rothe's method with respect to the time variable the non‐stationary problem is reduced to the sequence of boundary value problems for the non‐homogeneous Helmholtz equation. Further we construct the special integral representation for solutions and obtain the sequence of boundary integral equations (without volume integrals). For the full‐discretization of integral equations we propose some projection methods.     

Numerical solution of a nonuniquely solvable Volterra integral equation using extrapolation methods

In this work the numerical solution of a Volterra integral equation with a certain weakly singular kernel, depending on a real parameter μ, is considered. Although for certain values of μ this equation possesses an infinite set of solutions, we have been able to prove that Euler's method converges to a particular solution. It is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize, so that general extrapolation algorithms, like the E-algorithm, can be applied to improve the numerical results. This is illustrated by means of some examples.     

Fredholm积分方程特征值问题配置法外推的Matlab实验

通过Matlab编程作数值实验,探讨了Fredholm积分方程特征值问题配置法的Richardson外推规律,指出利用Richardson外推可以进一步提高近似特征值λh的精度阶.