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Existence theorems of solution to variational inequality problems

Science China mathematics 2001年第2期44卷 201-211页

作者：张立平韩继业徐大川 1. Institute of Applied Mathematics Academy of Mathematics and System Sciences Chinese Academy of Sciences 100080 Beijing China 2. Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences 100080 Beijing China

This paper introduces a new concept of exceptional family forvariational inequality problems with a general convex constrained set. By using this new concept, the authors establish a general sufficient condition for the existence of a solution to the problem. This condition is weaker than many known solution conditions and it is also necessary for pseudomonotone variational inequalities. Sufficient solution conditions for a class of nonlinear complementarity problems with P0 mappings are also obtained.

关键词： exceptional family variational inequality complementarity problem P0-function.

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A mixed finite element method for the unilateral contact problem in elasticity

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Science China mathematics 2006年第4期49卷 513-524页

作者： HUA Dongying & WANG Lieheng The First Fundamental Department, Beijing Information Technology institute, Beijing 100101, China institute of {2. mathematics and scientific/engineering computing, Academy of mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China 1. The First Fundamental Department Beijing Information Technology Institute Beijing 100101 China 2. Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100080 China

In this paper, we provide a new mixed finite element approximation of the variational inequality resulting from the unilateral contact problem in elasticity. We use the continuous piecewise P2.P1 finite element to approximate the displacement field and the normal stress component on the contact region. Optimal convergence rates are obtained under the reasonable regularity hypotheses. Numerical example verifies our results.

关键词： unilateral contact problem, mixed finite element method.

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The non-interior continuation methods for solving the function nonlinear complementarity problem

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Science China mathematics 2001年第9期44卷 1107-1114页

作者：黄正海韩继业徐大川张立平 1. Institute of Applied Mathematics Academy of Mathematics and System Sciences Chinese Academy o fSciences 100080 Beijing China 2. Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences 100080 Beijing China

In this paper, we propose a new smooth function that possesses a property not satisfied by the existing smooth functions. Based on this smooth function, we discuss the existence and continuity of the smoothing path for solving the P0 function nonlinear complementarity problem (NCP). Using the characteristics of the new smooth function, we investigate the boundedness of the iteration sequence generated by the non-interior continuation methods for solving the P0 function NCP under the assumption that the solution set of the NCP is nonempty and bounded. We show that the assumption that the solution set of the NCP is nonempty and bounded is weaker than those required by a few existing continuation methods for solving the NCP.

关键词： nonlinear complementarity problem non-interior continuation method P 0 function boundedness of the iteration sequence

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A new trust region algorithm for image restoration

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Science China mathematics 2005年第2期48卷 169-184页

作者： WEN Zaiwen & WANG Yanfei State Key Laboratory of scientific engineering {2., institute of computational mathematics and scientific/engineering {2., Academy of mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China National Key Laboratory on Remote Sensing Science, institute of Remote Sensing Applications, Chinese Academy of Sciences, Beijing 100101, China 1. State Key Laboratory of Scientific Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences 100080 Beijing China 2. National Key Laboratory on Remote Sensing Science Institute of Remote Sensing Applications Chinese Academy of Sciences 100101 Beijing China

The image restoration problems play an important role in remote sensing and astronomical image analysis. One common method for the recovery of a true image from corrupted or blurred image is the least squares error (LSE) method. But the LSE method is unstable in practical applications. A popular way to overcome instability is the Tikhonov regularization. However, difficulties will encounter when adjusting the so-called regularization parameter a. Moreover, how to truncate the iteration at appropriate steps is also challenging. In this paper we use the trust region method to deal with the image restoration problem, meanwhile, the trust region subproblem is solved by the truncated Lanczos method and the preconditioned truncated Lanczos method. We also develop a fast algorithm for evaluating the Kronecker matrix-vector product when the matrix is banded. The trust region method is very stable and robust, and it has the nice property of updating the trust region automatically. This releases us from tedious finding the regularization parameters and truncation levels. Some numerical tests on remotely sensed images are given to show that the trust region method is promising.

关键词： trust region algorithm, image restoration, Lanczos method, Kronecker matrix-vector product, preconditioning.

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Numerical simulation of Richtmyer-Meshkov instability

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Science China mathematics 2004年第z1期47卷 234-244页

作者： FU Dexun, MA Yanwen, ZHANG Linbo & TIAN BaolinState Key Laboratory of Nonlinear Mechanics, institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China State Key Laboratory of scientific and {2. computing, institute of computational mathematics, Chinese Academy of Sciences, Beijing 100080, China 1. State Key Laboratory of Nonlinear Mechanics Institute of Mechanics Chinese Academy of Sciences 100080 Beijing China2. State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics Chinese Academy of Sciences 100080 Beijing China

The compressible Navier-Stokes equations discretized with a fourth order accurate compact finite difference scheme with group velocity control are used to simulate the Richtmyer-Meshkov (R-M) instability problem produced by cylindrical shock-cylindrical material interface with shock Mach number Ms=1.2.and density ratio 1:2. (interior density/outer density). Effect of shock refraction, reflection, interaction of the reflected shock with the material interface, and effect of initial perturbation modes on R-M instability are investigated numerically. It is noted that the shock refraction is a main physical mechanism of the initial phase changing of the material surface. The multiple interactions of the reflected shock from the origin with the interface and the R-M instability near the material interface are the reason for formation of the spike-bubble structures. Different viscosities lead to different spike-bubble structure characteristics. The vortex pairing phenomenon is found in the initial double mode simulation. The mode interaction is the main factor of small structures production near the interface.

关键词： R-M instability, direct numerical simulation, shock-interface interaction.

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Convergence analysis of the adaptive finite element method with the red-green refinement

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Science China mathematics 2010年第2期53卷 499-512页

作者： ZHAO XuYing 1, 3 , HU Jun 2.? & SHI ZhongCi 1 1 State Key Laboratory of scientific and engineering {2., institute of computational mathematics and scientific/engineering {2., Academy of mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 2.Key Laboratory of mathematics and Applied mathematics (Peking University), Ministry of Education and School of Mathematical Sciences, Peking University, Beijing 100871, China 3 Graduate University of Chinese Academy of Sciences, Beijing 100190, China 1. State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China3. Graduate University of Chinese Academy of Sciences Beijing 100190 China2. Key Laboratory of Mathematics and Applied Mathematics (Peking University) Ministry of Education and School of Mathematical Sciences Peking University Beijing 100871 China

In this paper, we analyze the convergence of the adaptive conforming P 1 element method with the red-green refinement. Since the mesh after refining is not nested into the one before, the Galerkin-orthogonality does not hold for this case. To overcome such a difficulty, we prove some quasi-orthogonality instead under some mild condition on the initial mesh (Condition A). Consequently, we show convergence of the adaptive method by establishing the reduction of some total error. To weaken the condition on the initial mesh, we propose a modified red-green refinement and prove the convergence of the associated adaptive method under a much weaker condition on the initial mesh (Condition B).

关键词： red-green refinement adaptive finite element method convergence non-nested local refinement

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Preconditioners for higher order edge finite element discretizations of Maxwell's equations

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Science China mathematics 2008年第8期51卷 1537-1548页

作者： ZHONG LiuQiang1,2.SHU Shi1,2.SUN DuDu3&TAN Lin4 1School of Mathematical and computational Sciences,Xiangtan University,Xiangtan 411105,China 2.unan Key Laboratory for Computation and Simulation in Science and engineering,Xiangtan 411105,China 3{2. of computational mathematics and scientific/engineering computing,Academy of mathematicsand Systems Science,Graduate University of Chinese Academy of Sciences,Chinese Academy Sciences,P.O.Box 2.19,Beijing 100190,China 4Department of Math-Physics,Nanhua University,Hengyang 42.001,China 1. School of Mathematical and Computational Sciences Xiangtan University Xiangtan 411105 China 2. Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan 411105 China 3. Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Graduate University of Chinese Academy of Sciences Chinese Academy Sciences P.O. Box 2719 Beijing 100190 China 4. Department of Math-Physics Nanhua University Hengyang 421001 China

In this paper,we are concerned with the fast solvers for higher order edge finite element discretizations of Maxwell's *** present the preconditioners for the first family and second family of higher order N′ed′elec element equations,*** combining the stable decompositions of two kinds of edge finite element spaces with the abstract theory of auxiliary space preconditioning,we prove that the corresponding condition numbers of our preconditioners are uniformly bounded on quasi-uniform *** also present some numerical experiments to demonstrate the theoretical results.

关键词： preconditioner higher order edge finite element stable decomposition

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Two-step iterative methods for solving the stationary convection-diffusion equation with a small parameter at the highest derivative on a uniform grid

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computational mathematics and Mathematical Physics 2006年第2期46卷 282-293页

作者： Bai, Z.-Z. Krukier, L.A. Martynova, T.S. State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing 100080 P.O. Box 2719 China Computer Center Rostov State University Rostov-on-Don 344090 pr. Stachki 200/1 bld. 2 Russian Federation

A stationary convection-diffusion problem with a small parameter multiplying the highest derivative is considered. The problem is discretized on a uniform rectangular grid by the central-difference scheme. A new class of two-step iterative methods for solving this problem is proposed and investigated. The convergence of the methods is proved, optimal iterative methods are chosen, and the rate of convergence is estimated. Numerical results are presented that show the high efficiency of the methods. © MAIK "Nauka/Interperiodica" (Russia), 2.06.

关键词： Convergence Dissipative matrix Iterative methods Optimal iterative parameters Transition operator

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General liquid-crystal theory for anisotropically shaped molecules: Symmetry, orientational order parameters, and system free energy

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Physical Review E 2020年第6期102卷 062701-062701页

作者： Jie Xu Jeff Z. Y. Chen LSEC & NCMIS Institute of Computational Mathematics and Scientific/Engineering Computing (ICMSEC) Academy of Mathematics and Systems Science (AMSS) Chinese Academy of Sciences Beijing 100190 China Department of Physics and Astronomy University of Waterloo Ontario Canada N2L 3G1

A general theory of liquid crystals is presented, starting from the group-theory symmetry analysis of the constituting molecules. A particular attention is paid to the type of elastic free-energies and their relationships with the molecular symmetries. The orientational order-parameter tensors are identified for each molecular symmetry, in a consideration of consistently keeping the leading, characteristic elastic free energies in a model. The order parameters are expressed in terms of symmetric traceless tensors, some of high orders, for all major molecular symmetries, including seven groups of axial symmetries and seven groups of polyhedral symmetries. For spatially inhomogeneous liquid crystals, the couplings of these tensors in the elastic energies are derived by expanding the interaction energies between these molecules. The aim is to provide a general view of the molecular symmetries of individual molecules, orientational order parameters characterizing the orientational distribution functions, and the elastic free energies, all under one single group-theory approach.

关键词： Bent-core liquid crystals Liquid crystals Lyotropic liquid crystals Nematic liquid crystals

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An Absorbing-Layer Method of Seismic 3-D Migration for Irregular Survey Objects

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Chinese Journal of Geophysics 2013年第5期45卷 740-746页

作者： Yu-Xue WANG Guan-Quan ZHANG Rui-Fen ZHOU Yu-Ming ZHAO Department of Mathematics Daqing Petroleum Institute Daqing 151400 China Institute of Computational Mathematics and Scientific Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100080 China Department of Mechanical Engineering Daqing Petroleum Institute Daqing 151400 China No.2 Company of Drilling Daqing Petroleum Administration Brueau Daqing 163413 China

3-D seismic migration for an irregular acquisition geometry usually needs zero trace padding to form a regular area. The benefit of zero padding is easy to do 3-D migration, but it will reduce the computational efficiency and migration quality. In this paper we use an additional absorbing thin layer surrounding the irregular geometry instead of zero padding. It has greatly decreased computational cost and improved the migration quality. In order to form an absorbing layer, a damping factor is added to one-way wave equation. Since the wave field value attenuates quickly when the wave penetrates this thin layer, the reflection with the boundary condition that the wave-field value is zero is very small. Since the migration algorithm with the zero-boundary condition is easy to be implemented, 3-D migration for an irregular acquisition geometry can be done directly. The effectivity of this method is illustrated by processing the practical data.

关键词： Irregular acquisition geometry Migration Absorbing layer

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