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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

NONMONOTONE LOCAL MINIMAX METHODS FOR FINDING MULTIPLE SADDLE POINTS

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Journal of Computational Mathematics 2024年第3期42卷 851-884页

作者： Wei Liu ziqing xie Wenfan Yi South China Research Center for Applied Mathematics and Interdisciplinary Studies South China Normal UniversityGuangzhou 510631China Key Laboratory of Computing and Stochastic Mathematics(Ministry of Education) Hunan Normal UniversityChangshaHunan 410081China Hunan Provincial Key Laboratory of Intelligent Information Processing and Applied Mathematics School of MathematicsHunan UniversityChangshaHunan 410082China

In this paper,by designing a normalized nonmonotone search strategy with the BarzilaiBorwein-type step-size,a novel local minimax method(LMM),which is a globally convergent iterative method,is proposed and analyzed to find multiple(unstable)saddle points of nonconvex functionals in Hilbert *** to traditional LMMs with monotone search strategies,this approach,which does not require strict decrease of the objective functional value at each iterative step,is observed to converge faster with less ***,based on a normalized iterative scheme coupled with a local peak selection that pulls the iterative point back onto the solution submanifold,by generalizing the Zhang-Hager(ZH)search strategy in the optimization theory to the LMM framework,a kind of normalized ZH-type nonmonotone step-size search strategy is introduced,and then a novel nonmonotone LMM is *** feasibility and global convergence results are rigorously carried out under the relaxation of the monotonicity for the functional at the iterative ***,in order to speed up the convergence of the nonmonotone LMM,a globally convergent Barzilai-Borwein-type LMM(GBBLMM)is presented by explicitly constructing the Barzilai-Borwein-type step-size as a trial step-size of the normalized ZH-type nonmonotone step-size search strategy in each ***,the GBBLMM algorithm is implemented to find multiple unstable solutions of two classes of semilinear elliptic boundary value problems with variational structures:one is the semilinear elliptic equations with the homogeneous Dirichlet boundary condition and another is the linear elliptic equations with semilinear Neumann boundary *** numerical results indicate that our approach is very effective and speeds up the LMMs significantly.

关键词： Multiple saddle points Local minimax method Barzilai-Borwein gradient method Normalized nonmonotone search strategy Global convergence

A NUMERICAL STUDY OF UNIFORM SUPERCONVERGENCE OF LDG METHOD FOR SOLVING SINGULARLY PERTURBED PROBLEMS

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Journal of Computational Mathematics 2009年第2期27卷 280-298页

作者： ziqing xie Zuozheng Zhang Zhimin Zhang Key Laboratory of Computational and Stochastic Mathematics and Its Applications Universities of Hunan Province Hunan Normal University Changsha 410081 China

In this paper, we consider the local discontinuous Galerkin method （LDG） for solving singularly perturbed convection-diffusion problems in one- and two-dimensional settings. The existence and uniqueness of the LDG solutions are verified. Numerical experiments demonstrate that it seems impossible to obtain uniform superconvergence for numerical fluxes under uniform meshes. Thanks to the implementation of two-type different anisotropic meshes, i.e., the Shishkin and an improved grade meshes, the uniform 2p ＋ i-order superconvergence is observed numerically for both one-dimensional and twodimensional cases.

关键词： Singularly perturbed problems Local discontinuous Galerkin method Numerical fluxes Uniform superconvergence

Normalized Wolfe-Powell-type local minimax method for finding multiple unstable solutions of nonlinear elliptic PDEs

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Science China Mathematics 2023年第10期66卷 2361-2384页

作者： Wei Liu ziqing xie Wenfan Yi Key Laboratory of Computing and Stochastic Mathematics(Ministry of Education) Hunan Normal UniversityChangsha 410081China School of Mathematics Hunan UniversityChangsha 410082China

The local minimax method(LMM)proposed by Li and Zhou(2001,2002)is an efficient method to solve nonlinear elliptic partial differential equations(PDEs)with certain variational structures for multiple *** steepest descent direction and the Armijo-type step-size search rules are adopted in Li and Zhou(2002)and play a significant role in the performance and convergence analysis of traditional *** this paper,a new algorithm framework of the LMMs is established based on general descent directions and two normalized(strong)Wolfe-Powell-type step-size search *** corresponding algorithm framework,named the normalized Wolfe-Powell-type LMM(NWP-LMM),is introduced with its feasibility and global convergence rigorously justified for general descent *** a special case,the global convergence of the NWP-LMM combined with the preconditioned steepest descent(PSD)directions is also ***,it extends the framework of traditional *** addition,conjugate-gradient-type(CG-type)descent directions are utilized to speed up the ***,extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared with different algorithms in the LMM’s family to indicate the effectiveness and robustness of our *** practice,the NWP-LMM combined with the CG-type direction performs much better than its known LMM companions.

关键词： semilinear elliptic PDE multiple unstable solution local minimax method normalized strong Wolfe-Powell-type search rule conjugate-gradient-type descent direction general descent direction global convergence

SUPER-GEOMETRIC CONVERGENCE OF A SPECTRAL ELEMENT METHOD FOR EIGENVALUE PROBLEMS WITH JUMP COEFFICIENTS

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Journal of Computational Mathematics 2010年第3期28卷 418-428页

作者： Lin Wang ziqing xie Zhimin Zhang College of Mathematics and Computer Science Hunan Normal University Changsha 410081 China and College of Information Technical Science Nankai University Tianjin 300071 China College of Mathematics and Computer Science Hunan Normal University Changsha 310081 China Department of Mathematics Wayne State University Detroit MI 48202 USA and College of Mathematics and Computer Science Hunan Normal University Changsha 410081 China

We propose and analyze a C^0 spectral element method for a model eigenvalue problem with discontinuous coefficients in the one dimensional setting. A super-geometric rate of convergence is proved for the piecewise constant coefficients case and verified by numerical tests. Furthermore, the asymptotical equivalence between a Gauss-Lobatto collocation method and a spectral Galerkin method is established for a simplified model.

关键词： Eigenvalue Spectral method Collocation Galerkin finite element method.

Structure of multiple solutions for nonlinear differential equations

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Science China Mathematics 2004年第z1期47卷 172-180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,φjof -Δ, the multiple solutions for nonlinear problem Δu+f(u)=0 in Ω,u=0 onαΩ are approximated. A new search-extension method (SEM), which consists of three steps in three level subspaces, is proposed. Numerical simulations for several typical nonlinear cases, i.e. f(u)=u3,u2(u-p),u2(u2-p), are completed and some conjectures are presented.

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

Spectral Petrov-Galerkin Methods for the Second Kind Volterra Type Integro-Differential Equations

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Numerical Mathematics(Theory,Methods and Applications) 2011年第2期4卷 216-236页

作者： Xia Tao ziqing xie Xiaojun Zhou Key Laboratory of High Performance Computing and Stochastic Information Processing Ministry of Education of ChinaHunan Normal UniversityChangshaHunan 410081P.R.China School of Mathematics and Computer Science Guizhou Normal UniversityGuiyangGuizhou 550001P.R.China

This work is to provide general spectral and pseudo-spectral Jacobi-Petrov-Galerkin approaches for the second kind Volterra integro-differential *** Gauss-Legendre quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical *** some spectral and pseudo-spectral Jacobi-Petrov-Galerkin methods,a rigorous error analysis in both L2_(ω^(α,β))^(2),and L^(∞)norms is given provided that both the kernel function and the source function are sufficiently *** experiments validate the theoretical prediction.

关键词： Volterra integro-differential equation spectral Jacobi-Petrov-Galerkin pseudo-spectral Jacobi-Petrov-Galerkin spectral convergence

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.