2003年, 第21卷, 第6期　刊出日期：2003-11-15

  

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  DISCRETE MINUS ONE NORM LEAST-SQUARES FOR THE STRESS FORMULATION OF LINEAR ELASTICITY WITH NUMERICAL RESULTS$^*1)$

  Sang Dong Kim(1),Byeong Chun Shin(2),Seokchan Kim(3),Gyungsoo Woo(3)

  Journal of Computational Mathematics. 2003, 21(6): 689-702.

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  This paper studies the discrete minus one norm least-squares methods for the stress formulation of pure displacement linear elasticity in two dimensions. The proposed least-squares functional is defined as the sum of the $L^2-$ and $H^{-1}-$norms of the residual equations weighted appropriately. The minus one norm in the functional is replaced by the discrete minus one norm and then the discrete minus one norm least-squares methods are analyzed with various numerical results focusing on the finite element accuracy and multigrid convergence performances.
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  A NEW MULTI-SYMPLECTIC SCHEME FOR NONLINEAR

  Lang Yang HUANG(1),Wen Ping ZENG(1),Meng Zhao QIN(2)

  Journal of Computational Mathematics. 2003, 21(6): 703-714.

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  The Hamiltonian formulations of the linear 'good' Boussinesq (L.G.B.) equationn and the multi-symplectic formulation of the nonlinear "good" Boussinesq (N.G.B.) equation are considered. For the multi-symplectic formulation, a new fifteen-point difference scheme which is equivalent to the multi-symplectic Preissmann integrator is derived. We also present numerical experiments, which show that the symplectic and multi-symplectic schemes have excellent long-time numerical behavior.
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  DISIPATIVITY AND EXPONENTIAL STABILITY OF $\theta-$METHODS FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH A BOUNDED LAG$^*1)$

  Hong Jiong TIAN

  Journal of Computational Mathematics. 2003, 21(6): 715-726.

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  This paper deals with analytic and numerical dissipativity and exponential stability of singularly perturbed delay differential equations with any bounded state-independent lag. Sufficient conditions will be presented to ensurethat any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is dissipative and exponentially stable uniformly for sufficiently small $\epsilon>0$. We will study the numerical solution defined by the linear $\theta-$method and one-leg method and show that they are dissipative and exponentially stable uniformly for sufficiently small $\epsilon>0$ if and only if $\theta=1$.
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  SYMMETRIC POINT STRUCTURE OF SUPERCONVERGENCE FOR CUBIC TRIANGULAR ELEMENTS-A CONSULTATION WITH ZHU$*1)$

  Chuan Miao CHEN

  Journal of Computational Mathematics. 2003, 21(6): 727-732.

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  Superconvergence structrres for rectangular and triangular finite elements are summarized. Two debatable issues in Zhu's paper [18]are discussed. A superclose polynomisl to cubic triangular finite element is constructed by area coordinate.
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  THE DUAL MIXED METHOD FOR AN UNILATERAL PROBLEM $^*1)$

  Lie Heng Wang

  Journal of Computational Mathematics. 2003, 21(6): 733-746.

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  In this paper, the dual mixed method for an unilateral problem, which is the simplified modelling of scalar function for the friction-free contact problem, is considered. The dual mixed problem is introduced, the existence and uniqeness of the solution of the problem are presented, and error bounds $O(g^{\frac{3}{4}})$ and $O(g^{\frac{3}{2}})$ are obtained forthe dual mixed finite element approximations of Raviart-Thomas elements for $k=0$ and $k=1$ respectively.
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  A NEW SMOOTHING EQUATIONS QPPROACH TO THE NONLINEAR COMPLEMENTARITY PROBLEMS$^*1)$

  Chang Feng MA(1),Pu Yan NIE(2),Guo Ping LIANG(3)

  Journal of Computational Mathematics. 2003, 21(6): 747-758.

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  The nonlinear complementarity problem can be reformulated as a nonsmooth equation. In this paper we propose a new smoothing Newton algorithm for the solution of the nonlinear complementarity problem by construction a new smoothing approximation function. Global and local superlinear convergence results of the algorithm are obtained under suitable conditions. Numerical experiments confirm the good theoretical properties of the algorithm.
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  A TRUST REGION METHOD FOR SOLVING DISTRIBUTED PARAMETER IDENTIFICATION PROBLEMS$^*1)$

  Yan Fei WANG,Ya Xiang YUAN

  Journal of Computational Mathematics. 2003, 21(6): 759-772.

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  This paper is concerned with the ill-posed problems of identifying a parameter in an elliptic equation which appears in many applications in science and industry. Its solution is obtained by applying trust region method to a nonlinear least squares error problem. Trust region method gas long been a popular method for well-posed problems. This paper indicates that it is also suitable for ill-posed problems. Numerical experiment is given to compare the trust region method with the Tikhonov regularization method. It seems that the trust region method is more promising.
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  A CLASS OF ASYNCHRONOUS PARALLEL MULTISPLITTING RELAXATION METHODS FOR LARGE SPARSE LINEAR COMPLEMENTARITY PROBLEMS$^*1)$

  Zhong Zhi BAI(1),YU Guang HUANG(2)

  Journal of Computational Mathematics. 2003, 21(6): 773-790.

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  Asynchronous parallel multisplitting relaxation methods for solving large sparse linear complementarity problems are presented, and their convergence is proved when the system matrices are H-matrices having positive diagonal elements. Moreover, block and multi-parameter variants of the new methods, together with their convergence properties, are investigated in detail. Mumerical results show that these new methods can achieve high parallel efficiency for solving the large sparse linear complementarity problems on multiprocessor systems.
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  OPTIMALITY CONDITIONS OF A CLASS OF SPECIAL NONSMOOTH PROGRAMMING$^*1)$

  Song Bai SHENG(1),Hui Fu XU(2)

  Journal of Computational Mathematics. 2003, 21(6): 791-800.

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  In this paper, we investigate the optimality conditions of a class of special nonsmooth programming $\min F(x)=\sum_{i=1}^m|\max\{f_i(x),c_i\}|$ which arises from $L_1-$norm optimization, where $c_i\in R$ is constant and $f_i\in C^1,i=1,2,\cdots,m.$ these conditions can easily be tested by computer.
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  SOME NONLINEAR APPROXIMATIONS FOR MATRIX-VALUED FUNCTIONS$^*1)$

  Guo Liang XU

  Journal of Computational Mathematics. 2003, 21(6): 801-814.

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  Some nonlinear approximants, i.e., exponential-sum interpolation with equal distance or at origin, (0,1)-type fraction-sum approximations, for matrix-valued functions are introduced. All these approximation problems lead to a same form system of nonlinear equations. Solving methods for the nonlinear system are discussed. Conclusions on uniqueness and convergence of the approximants for certain class of functions are given.
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  A COMPUTATIONAL APPROACH FOR OPTIMAL CONTROL SYSTEMS GOVERNED BY PARABOLIC VARIATIONAL INEQUALITIES$^*$

  N.H.Sweilam,L.F.Abd-Elal

  Journal of Computational Mathematics. 2003, 21(6): 815-824.

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  Iterative techniques for solving optimal control systems governed by parabolic variational inequalities are presented. The techniques we use are based on linear finite elements method to approximate the state equations and nonlinear conjugate gradient methods to solve the discrete optimal control problem. Convergence results and numerical experiments are presented.
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  SUPERCONVERGENCE OF LEAST-SQUARES MIXED FINITE ELEMENT FOR SECOND-ORDER ELLIPTIC PROBLEMS$^*1)$

  Yan Ping CHEN(1),De Hao YU(2)

  Journal of Computational Mathematics. 2003, 21(6): 825-832.

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  In this paper the least-squares mixed finite element is considered for solving second-order elliptic problems in two dimensional domains. The primary solution u and the flux $\sigma$ are approximated using finite element spaces consisting of piecewise polynomials of degree k and r respectively. Based on interpolation operators and an auxiliary projection, superconvergent $H^1-$error estimates of both the primary solution approximation $u_h$and the flux approximation $\sigma_h$ are obtained under the standard quasi-uniform assumption on finite element partition. The superconvergence indicates an accuracy of $O(h^{r+2})$ for the least-squares mixed finite element approximation if Raviart-Thomas or Brezzi-Douglas-Fortin-Marini elements of order r are employed with optimal error estimate of $O(h^{r+1})$.