In this paper,we extend the work of Brenner and Sung[Math.Comp.59,321–338(1992)]and present a regularity estimate for the elastic equations in concave domains.Based on the regularity estimate we prove that the consta...In this paper,we extend the work of Brenner and Sung[Math.Comp.59,321–338(1992)]and present a regularity estimate for the elastic equations in concave domains.Based on the regularity estimate we prove that the constants in the error estimates of the nonconforming Crouzeix-Raviart element approximations for the elastic equations/eigenvalue problem are independent of Laméconstant,which means the nonconforming Crouzeix-Raviart element approximations are locking-free.We also establish two kinds of two-grid discretization schemes for the elastic eigenvalue problem,and analyze that when the mesh sizes of coarse grid and fine grid satisfy some relationship,the resulting solutions can achieve the optimal accuracy.Numerical examples are provided to show the efficiency of two-grid schemes for the elastic eigenvalue problem.展开更多
In this paper,based on the velocity-pressure formulation of the Stokes eigenvalue problemin d-dimensional case(d=2,3),we propose amultigrid discretization of discontinuous Galerkin method using P_(k)-P_(k)-1 element(k...In this paper,based on the velocity-pressure formulation of the Stokes eigenvalue problemin d-dimensional case(d=2,3),we propose amultigrid discretization of discontinuous Galerkin method using P_(k)-P_(k)-1 element(k≥1)and prove its a priori error estimate.We also give the a posteriori error estimators for approximate eigenpairs,prove their reliability and efficiency for eigenfunctions,and also analyze their reliability for eigenvalues.We implement adaptive calculation,and the numerical results confirm our theoretical predictions and show that our method is efficient and can achieve the optimal convergence order O(do f-2k/d).展开更多
A geometric intrinsic pre-processing algorithm(GPA for short)for solving largescale discrete mathematical-physical PDE in 2-D and 3-D case has been presented by Sun(in 2022–2023).Different from traditional preconditi...A geometric intrinsic pre-processing algorithm(GPA for short)for solving largescale discrete mathematical-physical PDE in 2-D and 3-D case has been presented by Sun(in 2022–2023).Different from traditional preconditioning,the authors apply the intrinsic geometric invariance,the Grid matrix G and the discrete PDE mass matrix B,stiff matrix A satisfies commutative operator BG=GB and AG=GA,where G satisfies G^(m)=I,m<<dim(G).A large scale system solvers can be replaced to a more smaller block-solver as a pretreatment in real or complex domain.In this paper,the authors expand their research to 2-D and 3-D mathematical physical equations over more wide polyhedron grids such as triangle,square,tetrahedron,cube,and so on.They give the general form of pre-processing matrix,theory and numerical test of GPA.The conclusion that“the parallelism of geometric mesh pre-transformation is mainly proportional to the number of faces of polyhedron”is obtained through research,and it is further found that“commutative of grid mesh matrix and mass matrix is an important basis for the feasibility and reliability of GPA algorithm”.展开更多
A novel neural network model, termed the discrete-time delayed standard neural network model （DDSNNM）, and similar to the nominal model in linear robust control theory, is suggested to facilitate the stability analy...A novel neural network model, termed the discrete-time delayed standard neural network model （DDSNNM）, and similar to the nominal model in linear robust control theory, is suggested to facilitate the stability analysis of discrete-time recurrent neural networks （RNNs） and to ease the synthesis of controllers for discrete-time nonlinear systems. The model is composed of a discrete-time linear dynamic system and a bounded static delayed （or non-delayed） nonlinear operator. By combining various Lyapunov functionals with the S-procedure, sufficient conditions for the global asymptotic stability and global exponential stability of the DDSNNM are derived, which are formulated as linear or nonlinear matrix inequalities. Most discrete-time delayed or non-delayed RNNs, or discrete-time neural-network-based nonlinear control systems can be transformed into the DDSNNMs for stability analysis and controller synthesis in a unified way. Two application examples are given where the DDSNNMs are employed to analyze the stability of the discrete-time cellular neural networks （CNNs） and to synthesize the neuro-controllers for the discrete-time nonlinear systems, respectively. Through these examples, it is demonstrated that the DDSNNM not only makes the stability analysis of the RNNs much easier, but also provides a new approach to the synthesis of the controllers for the nonlinear systems.展开更多
This study discusses generalized Rayleigh quotient and high efficiency finite element discretization schemes. Some results are as follows: 1) Rayleigh quotient accelerate technique is extended to nonselfadjoint proble...This study discusses generalized Rayleigh quotient and high efficiency finite element discretization schemes. Some results are as follows: 1) Rayleigh quotient accelerate technique is extended to nonselfadjoint problems. Generalized Rayleigh quotients of operator form and weak form are defined and the basic relationship between approximate eigenfunction and its generalized Rayleigh quotient is established. 2) New error estimates are obtained by replacing the ascent of exact eigenvalue with the ascent of finite element approximate eigenvalue. 3) Based on the work of Xu Jinchao and Zhou Aihui, finite element two-grid discretization schemes are established to solve nonselfadjoint elliptic differential operator eigenvalue problems and these schemes are used in both conforming finite element and non-conforming finite element. Besides, the efficiency of the schemes is proved by both theoretical analysis and numerical experiments. 4) Iterated Galerkin method, interpolated correction method and gradient recovery for selfadjoint elliptic differential operator eigenvalue problems are extended to nonselfadjoint elliptic differential operator eigenvalue problems.展开更多
This paper is a continuation of the author＇s paper in 2009, where the abstract theory of fold com- pleteness in Banach spaces has been presented. Using obtained there abstract results, we consider now very general bo...This paper is a continuation of the author＇s paper in 2009, where the abstract theory of fold com- pleteness in Banach spaces has been presented. Using obtained there abstract results, we consider now very general boundary value problems for ODEs and PDEs which poIynomially depend on the spectral parameter in both the equation and the boundary conditions. Moreover, equations and boundary conditions may con- rain abstract operators as well. So, we deal, generally, with integro-differential equations, functional-differential equations, nonlocal boundary conditions, multipoint boundary conditions, integro-differential boundary condi- tions. We prove n-fold completeness of a system of root functions of considered problems in the corresponding direct sum of Sobolev spaces in the Banach Lq-framework, in contrast to previously known results in the Hilbert L2-framework. Some concrete mechanical problems are also presented.展开更多
基金supported by the National Natural Science Foundation of China (Grant No.11761022)。
文摘In this paper,we extend the work of Brenner and Sung[Math.Comp.59,321–338(1992)]and present a regularity estimate for the elastic equations in concave domains.Based on the regularity estimate we prove that the constants in the error estimates of the nonconforming Crouzeix-Raviart element approximations for the elastic equations/eigenvalue problem are independent of Laméconstant,which means the nonconforming Crouzeix-Raviart element approximations are locking-free.We also establish two kinds of two-grid discretization schemes for the elastic eigenvalue problem,and analyze that when the mesh sizes of coarse grid and fine grid satisfy some relationship,the resulting solutions can achieve the optimal accuracy.Numerical examples are provided to show the efficiency of two-grid schemes for the elastic eigenvalue problem.
基金supported by the National Natural Science Foundation of China(Nos.12261024,11561014)the Science and Technology Planning Project of Guizhou Province(Guizhou Kehe fundamental research-ZK[2022]No.324).
文摘In this paper,based on the velocity-pressure formulation of the Stokes eigenvalue problemin d-dimensional case(d=2,3),we propose amultigrid discretization of discontinuous Galerkin method using P_(k)-P_(k)-1 element(k≥1)and prove its a priori error estimate.We also give the a posteriori error estimators for approximate eigenpairs,prove their reliability and efficiency for eigenfunctions,and also analyze their reliability for eigenvalues.We implement adaptive calculation,and the numerical results confirm our theoretical predictions and show that our method is efficient and can achieve the optimal convergence order O(do f-2k/d).
基金supported by the Basic Research Plan on High Performance Computing of Institute of Software(No.ISCAS-PYFX-202302)the National Key R&D Program of China(No.2020YFB1709502)the Advanced Space Propulsion Laboratory of BICE and Beijing Engineering Research Center of Efficient and Green Aerospace Propulsion Technology(No.Lab ASP-2019-03)。
文摘A geometric intrinsic pre-processing algorithm(GPA for short)for solving largescale discrete mathematical-physical PDE in 2-D and 3-D case has been presented by Sun(in 2022–2023).Different from traditional preconditioning,the authors apply the intrinsic geometric invariance,the Grid matrix G and the discrete PDE mass matrix B,stiff matrix A satisfies commutative operator BG=GB and AG=GA,where G satisfies G^(m)=I,m<<dim(G).A large scale system solvers can be replaced to a more smaller block-solver as a pretreatment in real or complex domain.In this paper,the authors expand their research to 2-D and 3-D mathematical physical equations over more wide polyhedron grids such as triangle,square,tetrahedron,cube,and so on.They give the general form of pre-processing matrix,theory and numerical test of GPA.The conclusion that“the parallelism of geometric mesh pre-transformation is mainly proportional to the number of faces of polyhedron”is obtained through research,and it is further found that“commutative of grid mesh matrix and mass matrix is an important basis for the feasibility and reliability of GPA algorithm”.
基金This work was supported by the National Natural Science Foundation of China （Grant No. 60504024） the Research Project of Zhejiang Provincial Education Department （Grant No. 20050905）.
文摘A novel neural network model, termed the discrete-time delayed standard neural network model （DDSNNM）, and similar to the nominal model in linear robust control theory, is suggested to facilitate the stability analysis of discrete-time recurrent neural networks （RNNs） and to ease the synthesis of controllers for discrete-time nonlinear systems. The model is composed of a discrete-time linear dynamic system and a bounded static delayed （or non-delayed） nonlinear operator. By combining various Lyapunov functionals with the S-procedure, sufficient conditions for the global asymptotic stability and global exponential stability of the DDSNNM are derived, which are formulated as linear or nonlinear matrix inequalities. Most discrete-time delayed or non-delayed RNNs, or discrete-time neural-network-based nonlinear control systems can be transformed into the DDSNNMs for stability analysis and controller synthesis in a unified way. Two application examples are given where the DDSNNMs are employed to analyze the stability of the discrete-time cellular neural networks （CNNs） and to synthesize the neuro-controllers for the discrete-time nonlinear systems, respectively. Through these examples, it is demonstrated that the DDSNNM not only makes the stability analysis of the RNNs much easier, but also provides a new approach to the synthesis of the controllers for the nonlinear systems.
基金supported by National Natural Science Foundation of China (Grant No.10761003) the Governor's Special Foundation of Guizhou Province for Outstanding Scientific Education Personnel (Grant No.[2005]155)
文摘This study discusses generalized Rayleigh quotient and high efficiency finite element discretization schemes. Some results are as follows: 1) Rayleigh quotient accelerate technique is extended to nonselfadjoint problems. Generalized Rayleigh quotients of operator form and weak form are defined and the basic relationship between approximate eigenfunction and its generalized Rayleigh quotient is established. 2) New error estimates are obtained by replacing the ascent of exact eigenvalue with the ascent of finite element approximate eigenvalue. 3) Based on the work of Xu Jinchao and Zhou Aihui, finite element two-grid discretization schemes are established to solve nonselfadjoint elliptic differential operator eigenvalue problems and these schemes are used in both conforming finite element and non-conforming finite element. Besides, the efficiency of the schemes is proved by both theoretical analysis and numerical experiments. 4) Iterated Galerkin method, interpolated correction method and gradient recovery for selfadjoint elliptic differential operator eigenvalue problems are extended to nonselfadjoint elliptic differential operator eigenvalue problems.
文摘This paper is a continuation of the author＇s paper in 2009, where the abstract theory of fold com- pleteness in Banach spaces has been presented. Using obtained there abstract results, we consider now very general boundary value problems for ODEs and PDEs which poIynomially depend on the spectral parameter in both the equation and the boundary conditions. Moreover, equations and boundary conditions may con- rain abstract operators as well. So, we deal, generally, with integro-differential equations, functional-differential equations, nonlocal boundary conditions, multipoint boundary conditions, integro-differential boundary condi- tions. We prove n-fold completeness of a system of root functions of considered problems in the corresponding direct sum of Sobolev spaces in the Banach Lq-framework, in contrast to previously known results in the Hilbert L2-framework. Some concrete mechanical problems are also presented.
