2003年, 第21卷, 第4期　刊出日期：2003-07-15

  

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  ROBUSTNESS OF AN UPWIND FINITE DIFFERENCE SCHEME FOR SEMILINEAR CONVECTION-DIFFUSION PROBLEMS WITH BOUNDARY TURNING POINTS

  Torsten LIN$\beta$

  Journal of Computational Mathematics. 2003, 21(4): 401-410.

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  We consider a singularly perturbed semilinear convection-diffusion problem with a boundary layer of attractive turning-point type. It is shown that its solution can be decomposed into a regular solution component and a layer component. This decomposition is used to analyse the convergence of an upwinded finite difference scheme on Shishkin meshes.
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  MULTIGRID FOR THE MORTAR FINITE ELEMENT FOR PARABOLIC PROBLEM

  Xue Jun XU(1),Jin Ru CHEN(2)

  Journal of Computational Mathematics. 2003, 21(4): 411-420.

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  In this paper, a mortar finite element method for parabolic problem is presented. Multi-grid method is used for solving the resulting discrete system. It is shown that the multigrid method is optimal, i.e, the convergence rate is independent of the mesh size L and the time step parameter $\tau$.
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  GAUSS-SEIDEL-TYPE MULTIGRID METHODS

  Zhao Hui HUANG,Qian Shun CHANG

  Journal of Computational Mathematics. 2003, 21(4): 421-434.

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  By making use of the Gauss-Seidel-type solution method, the procedure for computing the interpolation operator of multigrid methods is simplified. This leads to a saving of computational time. Three new kinds of interpolation formulae are obtained by adopting different approximate methods, to try to enhance the accuracy of the interpolatory operator. Atheoretical study proves the two-level convergence of these Gauss-Seidel-type MG methods. Aseries of numerical experiments is presented to evaluate the relative performance of the methods with respect to the convergence factor, CPU-time(for one V-cycle and the setup phase) and computational complexity.
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  THE GLOBAL ARTIFICIAL BOUNDARY CONDITIONS FOR NUMERICAL SIMULATIONS OF THE 3D FLOW AROUND A SUBMERGED BODY

  Hou De HAN,Xin WEN

  Journal of Computational Mathematics. 2003, 21(4): 435-450.

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  We consider the numerical approximations of the three-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface in a channel. One vertical side is introduced as the upstream artificial boundary and two vertical sies are introduced as the downstream artificial boundaries. On the artificial boundaries, a sequence of high-order global artificial boundary conditions are given. Then the original problem is reduced to a problem defined on a finite computational domain, which is equivalent to a variational problem. After solving the variational prlblem by the finite element method, we obtain the numerical approximation of the original problem. The numerical examples show that the artificial boundary conditions given in this paper are very effective.
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  PROXIMAL POINT ALGORITHM FOR MINIMIZATION OF DC FUNCTION

  Wen Yu SUN(1),Raimundo.J.B. Sampaio(2),M.A.B. Candido(2)

  Journal of Computational Mathematics. 2003, 21(4): 451-462.

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  In this paper we present some algorithms for minimization of DC function (difference of two convex functions). They are descent methods of the proximal-type which use the convex properties of the two convex functions separately. We also consider an approximate proximal point algorithm. Some properties of the $\epsilon$-subdifferential and the $\epsilon$-directional derivative are discussed. The convergence properties of the algorithms are established in both exact and approximate forms. Finally, we give some applications to the concave programming and maximum eigenvalue problems.
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  A SUCCESSIVE LEAST SQUARES METHOD FOR STRUCTURED TOTAL LEAST SQUARES

  Plamen Y. Yalamov(1),Jin Yun YUAN(2)

  Journal of Computational Mathematics. 2003, 21(4): 463-472.

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  A new method for Total Least Squares (TLS) problems is presented. It differs from previous approaches and is based on the solution of successive Least Squares problems. The method is quite suitable for Structured TLS (STLS) problems. We study mostly the casse of Toeplitz matrices in this paper. The numerical tests illustrate that the method converges to the solution fast for Toeplitz STLS problems. Since the method is designed for general TLS problems, other structured problems can be treated similarly.
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  REAL PIECEWISE ALGEBRAIC VARIETY

  Ren Hong WANG,Yi Sheng LAI

  Journal of Computational Mathematics. 2003, 21(4): 473-480.

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  We give definitions of real piecewise algebraic variety and its dimension. By using the techniques of real radical ideal, P-radical ideal, affine Hilbert polynomial, Bernstein-net form of polynomials on simplex, and decomposition of semi-algebraic set,etc.,we deal weth the dimension of the real pieciwise algebraic variety and real Nullstellensatz in $C^\mu$ spline ring.
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  ON THE ERROR ESTIMATE OF NONCONFORMING FINITE ELEMENT APPROXIMATION TO THE OBSTACLE PROBLEM

  Lie Heng WANG

  Journal of Computational Mathematics. 2003, 21(4): 481-490.

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  This paper is devoted to analysis of the nonconforming element approximation to the obstacle problem, and improvement and correction of the results in [11], [12].
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  FRACTIONAL DIFFUSION EQUATIONS WITH INTERNAL DEGREES OF FREEDOM

  Luis $V\acute{a}zquez$

  Journal of Computational Mathematics. 2003, 21(4): 491-494.

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  We present a generalization of the linear one-dimensional diffusion equation by combining the fractional derivatives and the internal degrees of freedom. The solutions are constructed from those of the scalar fractional diffusion equation. We analyze the interpolation between the standard diffusion process depending on the internal degrees of freedom associated to the system.
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  A MODIFIED VARIABLE-PENALTY ALTERNATING DIRECTIONS METHOD FOR MONOTONE VARIATIONAL INEQUALITIES

  Bing Sheng HE(1),Sheng Li WANG(1),Hai YANG(2)

  Journal of Computational Mathematics. 2003, 21(4): 495-504.

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  Alternating directions method is one of the approaches for solving linearly constrained separate monotone variational inequalities. Experience on applications has shown that the number of iteration significantly depends on the penalty for the system of linearly constrained equations and therefore the method with variable penalties is advantageous in practice. In this paper, we extend the Kontogiorgis and Meyer method [12] by removing the monotonicity assumption on the variable penalty matrices. <oreover, we introduce a self-adaptive rule that leads the method to be more efficient and insensitive for various initial penalties. Numerical results for a class of Fermat-Weber problems show that the modified method and its silf-adaptive technique are proper and necessary in practice.
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  THE INVERSE PROBLEM FOR PART SYMMETRIC MATRICES ON A SUBSPACE

  Zhen Yun PENG(1)，Xi Yan HU(2),Lei ZHANG(2)

  Journal of Computational Mathematics. 2003, 21(4): 505-512.

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  In this paper, the following two problems are considered:Problem Ⅰ. Given S∈Rn×p,X,B∈Rn×m, find A ∈ SRs,n such that AX = B, where SRs,n = {A∈ Rn×n|xT(A - AT) = 0, for all x ∈ R(S)}.Problem Ⅱ. Given A* ∈ Rn×n, find A ∈ SE such that ||$\hat{A}$-Given A* ||=minA ∈ SE ||A-Given A* ||, where SE is the solution set of Problem Ⅰ.The necessary and sufficient conditions for the solvability of and the general form of the solutions of problem I are given. For problem II, the expression for the solution, a numerical algorithm and a numerical example are provided.
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  TRAPEZOIDAL PLATE BENDING ELEMENT WITH DOUBLE SET PARAMETERS

  Shao Chun CHEN(1),Dong Yang SHI(1),I chro Hagixara(2)

  Journal of Computational Mathematics. 2003, 21(4): 513-518.

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  Using double set parameter method, a 12-parameter trapezoidal plate bending element is presented. The first set of degrees of freedom, which make the element convergent, are the values at the four vertices and the middle points of the four sides together with the mean values of the outer normal derivatives along four sides. The second set of degree of freedom, which make the number of unknowns in the resulting discrete system small and computation convenient are values and the first derivatives at the four vertices of the element.The convergence of the element is proved.
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  NUMERICAL DISSIPATION FOR THREE-POINT DIFFERENCE SCHEMES TO HYPERBOLIC EQUATIONS WITH UNEVEN MESHES

  Zi Niu WU

  Journal of Computational Mathematics. 2003, 21(4): 519-534.

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  The widely used locally adaptive Cartesian grid methods involve a series of abruptly refined interfaces. The numerical dissipation due to these interfaces is studied here for three-point difference approximations of a hyperbolic equation. It will be shown that if the wave moves in the fine-to-coarse direction then the dissipation is positive (stabilizing), and if the wave moves in the coarse-to-fine direction then the dissipation is negative (destabilizing).
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  DELAY-DEPENDENT TREATMENT OF LINEAR MULTISTEP METHODS FOR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

  Syed Khalid Jaffer(1),Ming Zhu LIU(2)

  Journal of Computational Mathematics. 2003, 21(4): 535-544.

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  This paper deals with a delay-dependent treatment of linear multistep methods for neutral delay differential equations y'(t) = ay(t) + by(t - τ) + cy'(t - τ), t > 0, y(t) = g(t), -τ≤ t ≤ 0, a,b andc ∈ R. The necessary condition for linear multistep methods to be $N_\tau(0)$-compatible. Figures of stability region for some linear multistep methods are depicted.