一类广义变分包含的迭代解

介绍一类新的涉及集值映射的变分包含问题，构造其迭代序列，并证明迭代序列收敛于变分包含问题的解，给出迭代序列与解的误差估计。     

最小二乘法及光滑问题不规则剖分上有限元解的超收敛分析

对有限元近似解提出一种通用的超收敛框架,该框架是对有限元解在另一有限维空间中作最小二乘逼近,中证明新构造的逼近解具有局部和整体上的超收敛,与所有已知的超收敛结果不同的是,该框架给出的超收敛结果对区域的有限元剖分没有附加任何一致性或对称性要求,这种得用小二乘作超收剑的技巧可以很简单地推广到混合有限元法,斯托克斯方程及重调方程的有限元法。     

含H-单调映象的完全广义非线性变分包含

引入和研究了一类新的完全广义非线性变分包含.在Hilbert空间中利用与H-单调映象相联系的预解算子的性质,对完全广义非线性变分包含建立了解的存在性定理和构造了一种新的迭代算法,证明了由此算法生成的迭代序列强收敛于精确解.其算法和结果是最近文献中相应算法和结果的改进和推广.     

Banach空间上变分不等式的一个超梯度方法

把王宜举等人[Modified extragradient—type method for variational inequali—ties and verification of the existence of solutions,J.Optim.Theory Appl.,2003,119:167-183]在欧氏空间上求解变分不等式的一个超梯度型方法推广到Banach空间.变分不等式中的算子不要求是一致连续的,其主要优点在于不管变分不等式是否有解,算法都是可执行的.此外,变分不等式的可解性可以通过算法产生的序列的性态来刻画.在适当的条件下,算法产生的序列强收敛于变分不等式的一个解,这是Bregman距离意义下离初始点最近的解.本文的主要结果推广和改善了近来文献中的相应结果.     

变分迭代法在抛物型方程反问题中的应用

本文将改进的变分迭代法的应用范围加以推广,使其应用于多维抛物型方程反问题中。它通过Lagrange乘子进行简便计算求得未知参量的精确值,再应用于多维抛物型方程反问题可以快速得到收敛于反问题精确解的收敛序列,从而得到精确解。同时,通过与Adomian’s分裂法结果比较可知前者比分裂法好。     

模糊映射的完全广义混合型强变分包含

研究一类新的关于模糊映射的完全广义混合型强变分包含问题，给出解的逼近算法，证明这类问题解的一个存在定理和序列收敛定理。     

对广义强非线性拟变分包含带有误差的近似点算法

本文研究了一类广义强非线性拟变分包含.在Hilbert空间内利用与极大单调映象相联系的预解算子的性质,对广义强非线性拟变分包含建立了解的存在性定理和建议了一个新的寻求近似解的带有误差的近似总算法,证明了近似解序列强收敛于精确解.作为特例,在此领域内的某些已知结果也被讨论.     

一类φ-强增生型变分包含解的存在和逼近问题

研究Banach空间中一类带有φ-强增生型映象的集值变分包含解的存在性及其具随机误差的Ishikawa迭代逼近问题,得到了迭代序列强收敛于变分包含问题的唯一解的若干等价条件,所得结果统一和推广了一些文献中的最新成果.     

模糊映象的完全广义变分包含

研究了一类新的关于模糊映射的完全广义变分包含问题 ,给出了解的逼近算法 ,证明了这类问题解的存在定理和序列收敛定理 ,推广了 N. J. Huang在文 [1 ]的主要结果 .     

无单调性集值变分不等式的一种投影算法

本文给出了求解无单调性集值变分不等式的一个新的投影算法,该算法所产生的迭代序列在Minty变分不等式解集非空且映射满足一定的连续性条件下收敛到解.对比文献[10]中的算法,本文中的算法使用了不同的线性搜索和半空间,在计算本文所引的两个数值例子时,该算法比文献[10]中的算法所需迭代步更少.     

基于二重网格的定常Navier-Stokes方程的局部和并行有限元算法

对二维定常的不可压缩的Navier-Stokes方程的局部和并行算法进行了研究．给出的算法是多重网格和区域分解相结合的算法,它是基于两个有限元空间:粗网格上的函数空间和子区域的细网格上的函数空间．局部算法是在粗网格上求一个非线性问题,然后在细网格上求一个线性问题,并舍掉内部边界附近的误差相对较大的解．最后,基于局部算法,通过有重叠的区域分解而构造了并行算法,并且做了算法的误差分析,得到了比标准有限元方法更好的误差估计,也对算法做了数值试验,数值结果通过比较验证了本算法的高效性和合理性．     

A normalized sparse linear equations solver

Normalized factorization procedures for the solution of large sparse linear finite element systems have been recently introduced in [3]. In these procedures the large sparse symmetric coefficient matrix of irregular structure is factorized exactly to yield a normalized direct solution method. Additionally, approximate factorization procedures yield implicit iterative methods for the finite difference or finite element solution. The numerical implementation of these algorithms is presented here and FORTRAN subroutines for the efficient solution of the resulting large sparse symmetric linear systems of algebraic equations are given.     

二阶椭圆问题的一类广义有限元法

本文提出了求解二阶椭圆问题的一类广义有限元方法,分析了广义有限元方法的优越性,证明了二阶椭圆问题的广义有限元方法具有比标准的Galerkin有限元方法更高阶的收敛速度,根据插值算子的性质,进一步证明了有限元解的亏量迭代校正收敛到广义有限元解,并用数值例子说明广义有限元方法是有效的.     

Local and parallel finite element algorithms based on two-grid discretizations for the transient Stokes equations

Based on two-grid discretizations, some local and parallel finite element algorithms for the d-dimensional (d = 2,3) transient Stokes equations are proposed and analyzed. Both semi- and fully discrete schemes are considered. With backward Euler scheme for the temporal discretization, the basic idea of the fully discrete finite element algorithms is to approximate the generalized Stokes equations using a coarse grid on the entire domain, then correct the resulted residue using a finer grid on overlapped subdomains by some local and parallel procedures at each time step. By the technical tool of local a priori estimate for the fully discrete finite element solution, errors of the corresponding solutions from these algorithms are estimated. Some numerical results are also given which show that the algorithms are highly efficient.     

Local and Parallel Finite Element Algorithms for Eigenvalue Problems

Abstract Some new local and parallel finite element algorithms are proposed and analyzed in this paper foreigenvalue problems.With these algorithms, the solution of an eigenvalue problem on a fine grid is reduced tothe solution of an eigenvalue problem on a relatively coarse grid together with solutions of some linear algebraicsystems on fine grid by using some local and parallel procedure.A theoretical tool for analyzing these algorithmsis some local error estimate that is also obtained in this paper for finite element approximations of eigenvectorson general shape-regular grids.     

A parallel solver for adaptive finite element discretizations

A parallel solver for the adaptive finite element analysis is presented. The primary aim of this work has been to establish an efficient parallel computational procedure which requires only local computations to update the solution of the system of equations arising from the finite element discretization after a local mesh-adaptation step. For this reason a set of algorithms has been developed (two-level domain decomposition, recursive hierarchical mesh-refinement, selective solution-update of linear systems of equations) which operate upon general and easily available partitioning, meshing and linear systems solving algorithms. AMS subject classification 15A23, 65N50, 65N60     

LOCAL AND PARALLEL FINITE ELEMENT ALGORITHMS FOR THE NAVIER-STOKES PROBLEM

Based on two-grid discretizations, in this paper, some new local and parallel finiteelement algorithms are proposed and analyzed for the stationary incompressible Navier-Stokes problem. These algorithms are motivated by the observation that for a solutionto the Navier-Stokes problem, low frequency components can be approximated well by arelatively coarse grid and high frequency components can be computed on a fine grid bysome local and parallel procedure. One major technical tool for the analysis is some locala priori error estimates that are also obtained in this paper for the finite element solutionson general shape-regular grids.     

Finite element potentials

We present an explicit and efficient way for constructing finite elements with assigned gradient, or curl, or divergence. Some simple notions of homology theory and graph theory applied to the finite element mesh are basic tools for devising the solution algorithms.     

Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs

A rigorous decomposition approach to solve separable mixed-integer nonlinear programs where the participating functions are nonconvex is presented. The proposed algorithms consist of solving an alternating sequence of Relaxed Master Problems (mixed-integer linear program) and two nonlinear programming problems (NLPs). A sequence of valid nondecreasing lower bounds and upper bounds is generated by the algorithms which converge in a finite number of iterations. A Primal Bounding Problem is introduced, which is a convex NLP solved at each iteration to derive valid outer approximations of the nonconvex functions in the continuous space. Two decomposition algorithms are presented in this work. On finite termination, the first yields the global solution to the original nonconvex MINLP and the second finds a rigorous bound to the global solution. Convergence and optimality properties, and refinement of the algorithms for efficient implementation are presented. Finally, numerical results are compared with currently available algorithms for example problems, illuminating the potential benefits of the proposed algorithm.     

Uzawa type algorithms for nonsymmetric saddle point problems

In this paper, we consider iterative algorithms of Uzawa type for solving linear nonsymmetric saddle point problems. Specifically, we consider systems, written as usual in block form, where the upper left block is an invertible linear operator with positive definite symmetric part. Such saddle point problems arise, for example, in certain finite element and finite difference discretizations of Navier-Stokes equations, Oseen equations, and mixed finite element discretization of second order convection-diffusion problems. We consider two algorithms, each of which utilizes a preconditioner for the operator in the upper left block. Convergence results for the algorithms are established in appropriate norms. The convergence of one of the algorithms is shown assuming only that the preconditioner is spectrally equivalent to the inverse of the symmetric part of the operator. The other algorithm is shown to converge provided that the preconditioner is a sufficiently accurate approximation of the inverse of the upper left block. Applications to the solution of steady-state Navier-Stokes equations are discussed, and, finally, the results of numerical experiments involving the algorithms are presented.