2003年, 第21卷, 第3期　刊出日期：2003-05-15

  

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  AN ITERATIVE HYBRIDIZED MIXED FINITE ELEMENT METHOD FOR ELLIPTIC INTERFACE PROBLEMS WITH STRONGLY DISCONTINUOUS COEFFICIENTS

  Dao Qi YANG(1),Jennifer ZHAO(2)

  Journal of Computational Mathematics. 2003, 21(3): 257-276.

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  An iterative algorithm is proposed and analyzed based on a hybridized mixed finite element method for numerically solving two-phase generalized Stefan interface problems with strongly discontinuous solutions, conormal derivatives, and coefficients. This algorithm iteratively solves small problems for each single phase with good accuracy and exchange information at the interface to advance the iteration until convergence, following the idea of Schwarz Alternating Methods. Error estimates are derived to show that this algorithm always converges provided that relaxation parameters are suitably chosen. Numeric experiments with matching and non-matching gtids at the interface from different phases are performed to show the qccuracy of the method for capturing discontinuities in the solutions and coefficients. In contrast to standard numerical methods, the accuracy of our method does not seem to deteriorate as the coefficient discontinuity increses.
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  MATHEMATICAL ANALYSIS FOR QUADRILATERAL ROTATED Q_1 ELEMENT Ⅱ: POINCARE INEQUALITY AND TRACE INEQUALITY

  Ping Bing MING,Zhong Ci SHI

  Journal of Computational Mathematics. 2003, 21(3): 277-286.

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  This is the second part of the paper for the mathematical study of nonconforming rotated Q1 element (NRQ1 hereafter) on arbitrary quadrilateral meshes. Some Poincare Inequalities are proved without assuming the quasi-uniformity of the mesh subdivision. A discrete trace inequality is also proved.
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  MATHEMATICAL ANALYSIS FOR QUADRILATERAL ROTATED Q_1 ELEMENT Ⅲ: THE EFFECT OF NUMERICAL INTEGRATION

  Ping Bing MING,Zhong Ci SHI

  Journal of Computational Mathematics. 2003, 21(3): 287-294.

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  This is the third part of the paper for the rotated Q1 nonconforming element on quadrilateral meshes for general second order elliptic problems. Some optimal numerical formulas are presented and analyzed. The novelty is that it includes a formula with only two sampling points which excludes even a Q1 unisolvent set. It is the optimal numerical integration formula over a quadrilateral mesh wigh least sampling points up to now.
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  CONIC TRUST REGION METHOD FOR LINEARLY CONSTRAINED OPTIMIZATION

  Wen Yu SUN(1),Jin Yun YUAN(2),Ya Xiang YUAN(3)

  Journal of Computational Mathematics. 2003, 21(3): 295-304.

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  In this paper we present a trust region method of conic model for linearly constrained optimization problems. We discuss trust region approaches with conic model subproblems. Some equivalent variation properties and optimality conditions are given. A trust region algorithm based on conic model is constructed. Global convergence of the method is established.
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  A NUMERICAL METHOD FOR DETERMINING THE OPTIMAL EXERCISE PRICE TO AMERICAN OPTIONS

  Xiong Hua WU,Xiu Juan FENG

  Journal of Computational Mathematics. 2003, 21(3): 305-310.

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  American options can be exercised prior to the date of expiration, the valuation of American options then constitutes a free boundary value problem. How to determine the free boundary, i.e. the optimal exercise price, is a key problem. In this paper, a nonlinear equation is given. The free boundary can be obtained by solving the nonlinear equation and the numerical results are better.
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  TESTING DIFFERENT CONJUGATE GRADIENT METHODS FOR LARGE-SCALE UNCONSTRAINED OPTIMIZATION

  Yu Hong DAI(1),Qin NI(2)

  Journal of Computational Mathematics. 2003, 21(3): 311-320.

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  In this paper we test different conjugate gradient (CG) methods for solving large-scale unconstrained optimization problems. The methods are divided in two groups: the first group includes five basic CG methods and the second five hybrid CG methods. A collection of medium-scale and large-scale test problems are drawn from a standard code of test problems, CUTE. The conjugate gradient methods are ranked according to the numerical results. Some remarks are given.
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  ON KORN'S INEQUALITY

  Lie Heng WANG

  Journal of Computational Mathematics. 2003, 21(3): 321-324.

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  This paper is devoted to give a new proof of Korn's inequality in LT - norm (1 < γ < ∞).
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  JACOBI SPECTRAL METHODS FOR MULTIPLE- DIMENSIONAL SINGULAR DIFFERENTIAL EQUATIONS

  Li Han WANG,Ben Yu GUO

  Journal of Computational Mathematics. 2003, 21(3): 325-338.

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  Jacobi polynomial approximations in multiple dimensions are investigated. They are applied to numerical solutions of singular differential equations. The convergence analysis and numerical results show their advantages.
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  THE PRIMAL-DUAL POTENTIAL REDUCTION ALGORITHM FOR POSITIVE SEMI-DEFINITE PROGRAMMING

  Si Ming HUANG

  Journal of Computational Mathematics. 2003, 21(3): 339-346.

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  In this paper we introduce a primal-dual potential reduction algorithm for positive semi-definite programming. Using the symetric preserving scalings for both primal and dual interior matrices, we can construct an algorithm which is very similar to the primaldual potential reduction algorithm of Huang and Kortanek [6] for linear programming. The complexity of the algorithm is either $O(n \log(X^0\cdot S^0/\epsilon))$ or $O(\sqrt{n}\log(X^0\cdot S^0/\epsilon))$ depends on the value of $\rho$ in the primal-dual potential function, where $X^0$ and $S^0$ is the initial interior matrices of the ppositive semi-definite programming.
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  A COMBINED HYBRID FINITE ELEMENT METHOD FOR PLATE BENDING PROBLEMS

  Tian Xiao ZHOU(1),Ziao Ping XIE(2)

  Journal of Computational Mathematics. 2003, 21(3): 347-356.

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  In this paper, a combined hybrid method is applied to finite element discretization of plate bending problems. It is shown that the resultant schemes are stabilized, i.e., the convergence of the schemes is independent of inf-sup conditions and any other patch test. Based on this, two new series of plate elements are proposed.
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  APPROXIMATION ALGORITHM FOR MAX-BISECTION PROBLEM WITH THE POSITIVE SEMIDEFINITE RELAXATION

  Da Chuan XU,Ji Ye HAN

  Journal of Computational Mathematics. 2003, 21(3): 357-366.

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  Using outward rotations, we obtain an approximation algorithm for Max-Bisection problem, i.e., partitioning the vertices of an undirected graph into two blocks of equal cardinality so as to maximize the weights of crossing edges. In many interesting cases, the algorithm performs better than the algorithms of Ye and of Halperin and Zwick, The main tool used to obtain this result is semidefinite programming.
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  PARALLEL DYNAMIC ITERATION METHODS FOR SOLVING NONLINEAR TIME-PERIODIC DIFFERENTIAL-ALGEBRAIC EQUATIONS

  Yao Lin JIANG

  Journal of Computational Mathematics. 2003, 21(3): 367-374.

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  In this paper we presented a convergence condition of parallel dynamic iteration methods for a nonlinear system of differential-algebraic equations with a periodic constraint. The convergence criterion is decided by the spectral expression of a linear operator derived from system partitions. Numerical experiments given here conirm the theoretical work of the paper.
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  STABILITIES OF (A,B,C) AND NPDIRK METHODS FOR SYSTEMS OF NEUTRAL DELAY-DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS

  Guo Feng ZHANG

  Journal of Computational Mathematics. 2003, 21(3): 375-382.

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  Consider the following neutral delay-differential equations with multiple delays (NMDDE)$$y'(t)=Ly(t)+\sum_{j=1}^{m}[M_jy(t-\tau_j)+N_jy'(t-\tau_j)],\ \ t\geq 0,\eqno(0.1)$$ where $\tau>0$, L, $M_j$ and $N_j$ are constant complex- value d×d matrices. A sufficient condition for the asymptotic stability of NMDDE system (0.1) is given. The stability of Butcher's (A,B,C)-method for systems of NMDDE are studied. In addition, we present a parallel diagonally-implicit iteration RK (PDIRK)methods (NPDIRK)for systems of NMDDE,which is easier to be implemented than fully implicit RK methos. We also investigate the stability of a special class of NPDIRK methods by analyzing their stability behaviors of the solutions of (0.1).
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  CONSERVATION OF THREE-POINT COMPACT SCHEMES ON SINGLE AND MULTIBLOCK PATCHED GRIDS FOR HYPERBOLIC PROBLEMS

  Zi Niu WU

  Journal of Computational Mathematics. 2003, 21(3): 383-400.

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  For nonlinear hyperbolic problems, conservation of the numerical scheme is important for convergence to the correct weak solutions. In this paper the conservation of the well-known compact scheme up to fourth order of accuracy on a single and uniform grid is studied, and a conservative interface treatment is derived for compact schemes on pathed grids. Gor a pure initial value problem, the compact scheme is shown to be equivalent to a scheme in the usual conservative form. For the case of a mixed initial boundary value problem, the compact scheme is conservative only if the rounding errors are small enough. For a patched grid interface, a conservative interface condition useful for mesh refinement and for parallel computation is derived and its order of local accuracy is analyzed.