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Science China Mathematics 2017年第7期60卷 1341-1352页

作者： chen chuanmiao HU HongLing Key Laboratory of HPCSIP(Ministry of Education of China) College of Mathematics and Computer ScienceHunan Normal University

To solve nonlinear system of equation,F(x) = 0,a continuous Newton flow x_t(t) = V(x) =-(DF(x))^(-1)F(x),x(0) =x^0 and its mathematical properties,such as the central field,global existence and uniqueness of real roots and the structure of the singular surface,are *** concisely introduce random Newton flow algorithm(NFA) for finding all roots,based on discrete Newton flow x^(j+1)=x^j+hV{x^j) with random initial value x^0 and h∈(0,1],and three computable quantities,g_j,d_j and K_*** numerical experiments with dimension n=300 are provided.

关键词： nonlinear system of equation continuous Newton flow real roots global existence and uniqueness random Newton flow algorithm

Extrapolation cascadic multigrid method on piecewise uniform grid

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Science China Mathematics 2013年第12期56卷 2711-2722页

作者： chen chuanmiao HU HongLing College of Mathematics and Computer Science Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China)Hunan Normal University

The triangular linear finite elements on piecewise uniform grid for an elliptic problem in convex polygonal domain are discussed. Global superconvergence in discrete Hi-norm and global extrapolation in discrete L2-norm are proved. Based on these global estimates the conjugate gradient method （CG） is effective, which is applied to extrapolation cascadic multigrid method （EXCMG）. The numerical experiments show that EXCMG is of the global higher accuracy for both function and gradient.

关键词： linear finite element piecewise uniform grid superconvergence extrapolation extrapolation cas-cadic multigrid method

Search-extension method for finding multiple solutions of nonlinear problem

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Science China Mathematics 2005年第6期48卷 721-734页

作者： chen chuanmiao XIE Ziqing Institute of Computation Hunan Normal University Changsha 410081 China Institute of Computation Hunan Normal University Changsha 410081 China

Based on the eigensystem {λj,φj}of -Δ, the multiple solutions for nonlinear problem Δu + f(u) = 0 in Ω, u = 0 on (?)Ω are approximated. A new search-extension method (SEM) is proposed, which consists of three algorithms in three level subspaces. Numerical experiments for f(u) = u3 in a square and L-shape domain are presented. The results show that there exist at least 3k - 1 distinct nonzero solutions corresponding to each κ-ple eigenvalue of -Δ (Conjecture 1).

关键词： nonlinear problem, multiple solutions, search-extension method, conjecture.

Analysis of search-extension method for finding multiple solutions of nonlinear problem

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Science China Mathematics 2008年第1期51卷 42-54页

作者： chen chuanmiao XIE ZiQing Institute of Computation Hunan Normal UniversityChangsha410081 China

For numerical computations of multiple solutions of the nonlinear elliptic problem Δu + f(u = 0 in Ω, u = 0 on Γ, a search-extension method (SEM) was proposed and systematically studied by the authors. This paper shall complete its theoretical analysis. It is assumed that the nonlinearity is non-convex and its solution is isolated, under some conditions the corresponding linearized problem has a unique solution. By use of the compactness of the solution family and the contradiction argument, in general conditions, the high order regularity of the solution u ∈ H 1+α, α > 0 is proved. Assume that some initial value searched by suitably many eigenbases is already fallen into the neighborhood of the isolated solution, then the optimal error estimates of its nonlinear finite element approximation are shown by the duality argument and continuation method.

关键词： nonlinear isolated solution regularity search-extension finite elements error analysis 65N30

FINITE ELEMENT METHOD WITH SUPERCONVERGENCE FOR NONLINEAR HAMILTONIAN SYSTEMS

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Journal of Computational Mathematics 2011年第2期29卷 167-184页

作者： chuanmiao chen Qiong Tang Shufang Hu College of Mathematics and Computer Science Hunan Normal University Changsha 410081 China Department of Information and Computation Hunan University of Technology Zhuzhou 412008 China

This paper is concerned with the finite element method for nonlinear Hamiltonian systems from three aspects： conservation of energy, symplicity, and the global error. To study the symplecticity of the finite element methods, we use an analytical method rather than the commonly used algebraic method. We prove optimal order of convergence at the nodes tn for mid-long time and demonstrate the symplecticity of high accuracy. The proofs depend strongly on superconvergence analysis. Numerical experiments show that the proposed method can preserve the energy very well and also can make the global trajectory error small for long time.

关键词： Nonlinear Hamiltonian systems Finiteelement method Superconvergence Energy conservation Symplecticity Trajectory.

An Extrapolation Cascadic MultigridMethod for Elliptic Problems on Reentrant Domains

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Advances in Applied Mathematics and Mechanics 2017年第6期9卷 1347-1363页

作者： Kejia Pan Dongdong He chuanmiao chen School of Mathematics and Statistics Central South UniversityChangshaHunan 410083China School of Aerospace Engineering and Applied Mechanics Tongji UniversityShanghai 200092China College of Mathematics and Computer Science Key Laboratory of High Performance Computing and Stochastic Information Processing(Ministry of Education of China)Hunan Normal UniversityChangshaHunan 410081China

This paper proposes an extrapolation cascadic multigrid(EXCMG)method to solve elliptic problems in domains with reentrant *** a class ofλ-graded meshes,we derive some new extrapolation formulas to construct a high-order approximation to the finite element solution on the next finer mesh using the numerical solutions on two-level of grids(current and previous grids).Then,this high-order approximation is used as the initial guess to reduce computational cost of the conjugate gradient *** application of this idea results in the EXCMG method proposed in this ***,numerical results for a crack problem and an L-shaped problem are presented to verify the efficiency and effectiveness of the proposed EXCMG method.

关键词： Richardson extrapolation Cascadicmultigrid gradedmesh elliptic problems corner singularity

Implicit DG Method for Time Domain Maxwell’s Equations Involving Metamaterials

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Advances in Applied Mathematics and Mechanics 2015年第6期7卷 796-817页

作者： Jiangxing Wang Ziqing Xie chuanmiao chen Key Laboratory of High Performance Computing and Stochastic Information Processing College of Mathematics and Computer ScienceHunan Normal UniversityChangsha 410081China

An implicit discontinuous Galerkin method is introduced to solve the timedomain Maxwell’s equations in *** Maxwell’s equations in metamaterials are represented by integral-differential *** scheme is based on discontinuous Galerkin method in spatial domain and Crank-Nicolson method in temporal *** fully discrete numerical scheme is proved to be unconditionally *** polynomial of degree at most p is used for spatial approximation,our scheme is verified to converge at a rate of O(τ^(2)+h^(p)+1/2).Numerical results in both 2D and 3D are provided to validate our theoretical prediction.

关键词： Maxwell’s equations metamaterials fully disctete DG method L2-stability L2-error estimate.

TIME-EXTRAPOLATION ALGORITHM （TEA） FOR LINEAR PARABOLIC PROBLEMS

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Journal of Computational Mathematics 2014年第2期32卷 183-194页

作者： Hongling Hu chuanmiao chen Kejia Pan College of Mathematics and Computer Science Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China) Hunan Normal University Changsha 410081 China School of Mathematics and Statistics Central South University Changsha 410083 China

The fast solutions of Crank-Nicolson scheme on quasi-uniform mesh for parabolic prob- lems are discussed. First, to decrease regularity requirements of solutions, some new error estimates are proved. Second, we analyze the two characteristics of parabolic discrete scheme, and find that the efficiency of Multigrid Method （MG） is greatly reduced. Nu- merical experiments compare the efficiency of Direct Conjugate Gradient Method （DCG） and Extrapolation Cascadic Multigrid Method （EXCMG）. Last, we propose a Time- Extrapolation Algorithm （TEA）, which takes a linear combination of previous several level solutions as good initial values to accelerate the rate of convergence. Some typical extrapolation formulas are compared numerically. And we find that under certain accuracy requirement, the CG iteration count for the 3-order and 7-level extrapolation formula is about 1/3 of that of DCG＇s. Since the TEA algorithm is independent of the space dimension, it is still valid for quasi-uniform meshes. As only the finest grid is needed, the proposed method is regarded very effective for nonlinear parabolic problems.

关键词： Parabolic problem Crank-Nicolson scheme Error estimates Time-extrapolationalgorithm CG-iteration.

ON EXTRAPOLATION CASCADIC MULTIGRID METHOD

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Journal of Computational Mathematics 2011年第6期29卷 684-697页

作者： chuanmiao chen Zhong-Ci Shi Hongling Hu Institute of Computation Hunan Normal University Changsha 410081 China LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China

Based on an asymptotic expansion of （bi）linear finite elements, a new extrapolation formula and extrapolation cascadic multigrid method （EXCMG） are proposed. The key ingredients of the proposed methods are some new extrapolations and quadratic interpolations, which are used to provide better initial values on the refined grid. In the case of triple grids, the errors of the new initial values are analyzed in detail. The numerical experiments show that EXCMG has higher accuracy and efficiency.

关键词： Cascadic multigrid Finite element New extrapolation Error analysis.

Asymptotic expansions of finite element solutions to Robin problems in H^3 and their application in extrapolation cascadic multigrid method

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Science China Mathematics 2014年第4期57卷 687-698页

作者： HU HongLing chen chuanmiao PAN KeJia College of Mathematics and Computer Science Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China)Hunan Normal University School of Mathematics and Statistics Central South University Key Laboratory of Metallogenic Prediction of Nonferrous Metals Ministry of EducationSchool of Geosciences and Info-PhysicsCentral South University

For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the asymptotic error expansions of bilinear finite element have the accuracy of O(h3)for u∈*** on the obtained asymptotic error expansions for linear finite elements,extrapolation cascadic multigrid method(EXCMG)can be used to solve Robin problems ***,by virtue of Richardson not only the accuracy of the approximation is improved,but also a posteriori error estimation is ***,some numerical experiments that confirm the theoretical analysis are presented.

关键词： finite element Richardson extrapolation Robin problem asymptotic expansion cascadic multi-grid method