可压缩流动的Fourier谱-有限元解法

本文考虑n维(n=2,3)可压缩流动的带有单向周期边值条件问题的数值解.我们在周期方向采用Fourier谱方法,在非周期方向采用有限元方法,从而构造了一类谱-有限元格式.文中严格分析了计算误差,得到了收敛阶的估计.     

双偶次有限元的渐近准确误差估计

当我们用有限元方法近似求解偏微分方程的边值问题时,常对近似解有一定的精度要求.于是仅在初始网格上进行一次计算是不够的,往往要进行一系列的计算.如何根据对已有计算结果的分析来控制下一步计算,导致自适应方法的出现.自适应方法的基础是对有限元近似解作后验误差估计.在h型自适应有限元方法中,通过加细剖分来达     

Stokes问题有限元逼近中求导数的提取法

1.引言 我们知道Poisson方程和平面弹性问题的解的导数的近似值可以通过所谓提取公式得到,而不必对近似解直接求导数.这样我们可以得到具有与近似解本身同阶精度的导数的近似值.这一方法已被用于基于插值误差的后验误差估计及相应的自适应有限元方法中本文将这一方法应用于Stokes问题的有限元逼近,从Stokes方程的解的     

抛物型方程基于POD方法的有限元格式

将特征正交分解(Proper Orthogonal Decomposition, 简记为POD)方法应用于抛物型方程通常的有限元格式, 简化其为一个计算量很少但 具有足够高精度的POD有限元格式, 并给出简化的POD有限元解的误差分析. 数值例子表明在简化的POD有限元解和通常的有限元解之间的误差足够小的情形下, POD有限元格式比通常的有限元格式大大地节省计算量, 从而验证POD方法的有效性.     

求积元法在金融工程计算领域应用初探

金融工程领域的大量实际问题最终都可归结为对随机微分方程(组)的求解.针对金融工程计算领域涉及到的静态一维问题,首次将求积元方法应用于非自伴随微分方程的求解.建立了相应的求积元方法计算单元.对典型问题进行计算,并与解析解、有限差分解、有限元解分别进行对比.结果表明,求积元法是一种简单准确高效的数值方法,可进一步用于金融工程计算领域动态问题、二维问题的计算分析.     

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     

基于POD方法带Poisson跳的随机扩散流行病模型数值模拟

将特征正交分解(proper orthogonal decomposition,简记为POD)方法结合有限元方法应用于带Poisson跳的扩散流行病模型,简化其为一个具有较低维数和较高精度的有限元格式,并给出POD有限元解和通常有限元解的误差分析.数值例子表明在POD有限元降维解和通常有限元解之间的误差足够小的情况下,POD有限元方法能大大地降低维数,提高计算速度和计算精度,从而验证带Poisson跳的随机扩散流行病模型的POD有限元格式是可行和有效的.     

一类组合结构平衡问题与特征值问题的非协调有限元方法

利用有限元方法求解弹性组合结构问题,往往遇到非协调情形.这不仅由于各构件采用的有限元本身的非协调性,而且在于各构件有限元间的连接条件一般不相匹配.在[2]中.我们曾利用Strang的结果,对板-梁组合结构一些具体的非协调有限元解进行     

奇异解的校正

一、主要结果文[1]以渐近展开式为基本工具,研究了用线性有限元解的高次插值构造亏量的有限元亏量校正方法,这是有限元高精度算法的重大进展.但文[1]对解的光滑性作了较强的假设,对一些问题,例如凹角域问题,解具有奇异性,文[1]的论证方法无效.本文以超收敛为基本工具,用不同于文[1]的论证方法证明了有限元校正方法可提高凹角域问题有限元解的精度阶.     

周期结构区域中压电问题的双尺度有限元方法

近年来纤维压电复合材料的力电性能预测已发展为一个重要的研究领域．对力电耦合周期结构的复合材料问题,通过引入匹配的边界层得到了电势与位移解的新型双尺度有限元计算方法,建立了电势与位移的双尺度耦合关系,分析了双尺度有限元解的误差.数值算例验证了方法的有效性．     

系统数值优化方法的研究与应用

大系统具有结构复杂、功能综合、因素众多等特点,相应的规划和决策问题的求解是非常困难的。我们针对大系统工程规划和决策的特点,结合运筹学的原理,提出了大系统数值优化概念,建立了优化模型,给出了优化问题的求解方法;并选择了中石油的DS-1油田作为应用实例,进行了油田采油工程数值优化,取得了很好的效果。为一般的复杂工程系统的规划和决策提供了一个新方法。     

An experimental analysis of a population based approach for global optimization

In this paper we perform a computational analysis of a population based approach for global optimization, Population Basin Hopping (PBH), which was proven to be very efficient on very challenging global optimization problems by the authors (see ). The experimental analysis aims at understanding more deeply how the approach works and why it is successful on challenging problems.     

A primal majorized semismooth Newton-CG augmented Lagrangian method for large-scale linearly constrained convex programming

In this paper, we propose a primal majorized semismooth Newton-CG augmented Lagrangian method for large-scale linearly constrained convex programming problems, especially for some difficult problems. The basic idea of this method is to apply the majorized semismooth Newton-CG augmented Lagrangian method to the primal convex problem. And we take two special nonlinear semidefinite programming problems as examples to illustrate the algorithm. Furthermore, we establish the global convergence and the iteration complexity of the algorithm. Numerical experiments demonstrate that our method works very well for the testing problems, especially for many ill-conditioned ones.     

Robust Stability and Performance Analysis of Uncertain Systems Using Linear Matrix Inequalities

A wide variety of problems in system and control theory can be formulated or reformulated as convex optimization problems involving linear matrix inequalities (LMIs), that is, constraints requiring an affine combination of symmetric matrices to be positive semidefinite. For a few very special cases, there are analytical solutions to these problems, but in general LMI problems can be solved numerically in a very efficient way. Thus, the reduction of a control problem to an optimization problem based on LMIs constitutes, in a sense, a solution to the original problem. The objective of this article is to provide a tutorial on the application of optimization based on LMIs to robust control problems. In the first part of the article, we provide a brief introduction to optimization based on LMIs. In the second part, we describe a specific example, that of the robust stability and performance analysis of uncertain systems, using LMI optimization.     

Splitting-based conjugate gradient method for a multi-dimensional linear inverse heat conduction problem

In this paper we consider a multi-dimensional inverse heat conduction problem with time-dependent coefficients in a box, which is well-known to be severely ill-posed, by a variational method. The gradient of the functional to be minimized is obtained by the aid of an adjoint problem, and the conjugate gradient method with a stopping rule is then applied to this ill-posed optimization problem. To enhance the stability and the accuracy of the numerical solution to the problem, we apply this scheme to the discretized inverse problem rather than to the continuous one. The difficulties with large dimensions of discretized problems are overcome by a splitting method which only requires the solution of easy-to-solve one-dimensional problems. The numerical results provided by our method are very good and the techniques seem to be very promising.     

Coincidence and maximal element theorems and their applications to generalized equilibrium problems and minimax inequalities

In this paper we obtain first a very general coincidence theorem. From this we derive a new coincidence theorem and two alternative theorems concerning existence of maximal elements. Applications of these results to generalized equilibrium problems and minimax inequalities are given in the last sections.     

New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems

In this article we present a new fixed point theorem for a class of general mixed monotone operators, which extends the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for mixed monotone operators. Based on them the local existence-uniqueness of positive solutions for nonlinear boundary value problems which include Neumann boundary value problems, three-point boundary value problems and elliptic boundary value problems for Lane-Emden-Fowler equations is proved. The theorems for nonlinear boundary value problems obtained here are very general.     

Regularized mixed quasi equilibrium problems

In this paper, we introduce and study a new class of equilibrium problems, known as regularized mixed quasi equilibrium problems. We use the auxiliary principle technique to suggest and analyze some iterative schemes for regularized equilibrium problems. We prove that the convergence of these iterative methods requires either pseudomonotonicity or partially relaxed strongly monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier results for solving equilibrium problems and variational inequalities involving the convex sets.     

A method to bypass the lack of solutions in MinSup problems under quasi-equilibrium constraints

MinSup problems with constraints described by quasi-equilibrium problems are considered in Banach spaces. The solutions set of such problems may be empty even in very good situations, so the aim of this paper is twofold. First, we determine appropriate regularizations (called inner regularizations) which allow to reach the value of the original problem. Then, among these regularizations we identify those which allow to bypass the lack of exact solutions to these problems by a suitable concept of “viscosity” solution whose existence is then proved under reasonable assumptions.     

Some numerical experiments with multigrid methods on Shishkin meshes

Piecewise uniform meshes introduced by Shishkin, are a very useful tool to construct robust and efficient numerical methods to approximate the solution of singularly perturbed problems. For small values of the diffusion coefficient, the step size ratios, in this kind of grids, can be very large. In this case, standard multigrid methods are not convergent. To avoid this troublesome, in this paper we propose a modified multigrid algorithm, which works fine on Shishkin meshes. We show some numerical experiments confirming that the proposed multigrid method is convergent, and it has similar properties that standard multigrid for classical elliptic problems.