This paper focuses on the analytical and numerical asymptotical stability of neutral reaction-diffusion equations with piecewise continuous arguments.First,for the analytical solutions of the equations,we derive their...This paper focuses on the analytical and numerical asymptotical stability of neutral reaction-diffusion equations with piecewise continuous arguments.First,for the analytical solutions of the equations,we derive their expressions and asymptotical stability criteria.Second,for the semi-discrete and one-parameter fully-discrete finite element methods solving the above equations,we work out the sufficient conditions for assuring that the finite element solutions are asymptotically stable.Finally,with a typical example with numerical experiments,we illustrate the applicability of the obtained theoretical results.展开更多
In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the m...In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.展开更多
The transient behavior of a semiconductor device consists of a Poisson equation for the electric potential and of two nonlinear parabolic equations for the electron density and hole density. The electric potential equ...The transient behavior of a semiconductor device consists of a Poisson equation for the electric potential and of two nonlinear parabolic equations for the electron density and hole density. The electric potential equation is discretized by a mixed finite element method. The electron and hole density equations are treated by implicit-explicit multistep finite element methods. The schemes are very efficient. The optimal order error estimates both in time and space are derived.展开更多
In this paper, we represent a new numerical method for solving the nonstationary Stokes equations in an unbounded domain. The technique consists in coupling the boundary integral and finite element methods. The variat...In this paper, we represent a new numerical method for solving the nonstationary Stokes equations in an unbounded domain. The technique consists in coupling the boundary integral and finite element methods. The variational formulation and well posedness of the coupling method are obtained. The convergence and optimal estimates for the approximation solution are provided.展开更多
By applying the continuous finite element methods of ordinary differential equations, the linear element methods are proved having second-order pseudo-symplectic scheme and the quadratic element methods are proved hav...By applying the continuous finite element methods of ordinary differential equations, the linear element methods are proved having second-order pseudo-symplectic scheme and the quadratic element methods are proved having third-order pseudo- symplectic scheme respectively for general Hamiltonian systems, and they both keep energy conservative. The finite element methods are proved to be symplectic as well as energy conservative for linear Hamiltonian systems. The numerical results are in agree-ment with theory.展开更多
The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type,coupled through their nonlinear terms.In our previous work[9],we constructed conservative and dissipative finite ...The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type,coupled through their nonlinear terms.In our previous work[9],we constructed conservative and dissipative finite element methods for these systems and presented a priori error estimates for the semidiscrete schemes.In this sequel,we present a posteriori error estimates for the semidiscrete and fully discrete approximations introduced in[9].The key tool employed to effect our analysis is the dispersive reconstruction devel-oped by Karakashian and Makridakis[20]for related discontinuous Galerkin methods.We conclude by providing a set of numerical experiments designed to validate the a posteriori theory and explore the effectivity of the resulting error indicators.展开更多
On the basis of the concept of finite element methods, the rigorous analytical solutions of structural response in terms of the design variables are researched in this paper. The spatial trusses are taken as an exampl...On the basis of the concept of finite element methods, the rigorous analytical solutions of structural response in terms of the design variables are researched in this paper. The spatial trusses are taken as an example for the solution of the analytical expressions of the explicit displacements which are proved mathematically; then some conclusions are reached that are useful to structural sensitivity analysis and optimization. In the third part of the paper, a generalized geometric programming method is sugguested for the optimal model with the explicit displacement. Finally, the analytical solutions of the displacements of three trusses are given as examples.展开更多
Magnetization configurations were calculated under various magnetic fields for nanocrystalline Pr-Fe-B permanent magnets by micromagnetic finite element method.According to the configurations during demagnetization pr...Magnetization configurations were calculated under various magnetic fields for nanocrystalline Pr-Fe-B permanent magnets by micromagnetic finite element method.According to the configurations during demagnetization process, the mechanism of magnetization reversal was analyzed.For the Pr2Fe14B with 10 nm grains or its composite with 10vol.% α-Fe, the coercivity was determined by nucleation of reversed domain that took place at grain boundaries.However, for Pr2Fe14B with 30 nm grains, coercivity was controlled by pinning of the nucle-ated domain.For Pr2Fe14B/α-Fe with 30vol.% α-Fe, the demagnetization behavior was characterized by continuous reversal of α-Fe moment.展开更多
In this paper,we consider the numerical schemes for a timefractionalOldroyd-B fluidmodel involving the Caputo derivative.We propose two efficient finite element methods by applying the convolution quadrature in time g...In this paper,we consider the numerical schemes for a timefractionalOldroyd-B fluidmodel involving the Caputo derivative.We propose two efficient finite element methods by applying the convolution quadrature in time generated by the backward Euler and the second-order backward difference methods.Error estimates in terms of data regularity are established for both the semidiscrete and fully discrete schemes.Numerical examples for two-dimensional problems further confirmthe robustness of the schemes with first-and second-order accurate in time.展开更多
In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique ...In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.展开更多
Chaotic vibrations of flexible non-linear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions(symmetric and non-symmetric)are studied in this work.Reliability of the obtained result...Chaotic vibrations of flexible non-linear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions(symmetric and non-symmetric)are studied in this work.Reliability of the obtained results is verified by the finite difference method(FDM)and the finite element method(FEM)with the Bubnov-Galerkin approximation for various boundary conditions and various dynamic regimes(regular and non-regular).The influence of boundary conditions on the Euler-Bernoulli beams dynamics is studied mainly,dynamic behavior vs.control parameters { ωp,q0 } is reported,and scenarios of the system transition into chaos are illustrated.展开更多
On triangle or quadrilateral meshes, two finite element methods are proposed for solving the Reissner-Mindlin plate problem either by augmenting the Galerkin formulation or modifying the plate-thickness. In these meth...On triangle or quadrilateral meshes, two finite element methods are proposed for solving the Reissner-Mindlin plate problem either by augmenting the Galerkin formulation or modifying the plate-thickness. In these methods, the transverse displacement is approximated by conforming (bi)linear macroelements or (bi)quadratic elements, and the rotation by conforming (bi)linear elements. The shear stress can be locally computed from transverse displacement and rotation. Uniform in plate thickness, optimal error bounds are obtained for the transverse displacement, rotation, and shear stress in their natural norms. Numerical results are presented to illustrate the theoretical results.展开更多
In this article,two kinds of expandable parallel finite element methods,based on two-grid discretizations,are given to solve the linear elliptic problems.Compared with the classical local and parallel finite element m...In this article,two kinds of expandable parallel finite element methods,based on two-grid discretizations,are given to solve the linear elliptic problems.Compared with the classical local and parallel finite element methods,there are two attractive features of the methods shown in this article:1)a partition of unity is used to generate a series of local and independent subproblems to guarantee the final approximation globally continuous;2)the computational domain of each local subproblem is contained in a ball with radius of O(H)(H is the coarse mesh parameter),which means methods in this article are more suitable for parallel computing in a large parallel computer system.Some a priori error estimation are obtained and optimal error bounds in both H^1-normal and L^2-normal are derived.Finally,numerical results are reported to test and verify the feasibility and validity of our methods.展开更多
The mixed finite element(MFE) methods for a shallow water equation system consisting of water dynamics equations,silt transport equation,and the equation of bottom topography change were derived.A fully discrete MFE s...The mixed finite element(MFE) methods for a shallow water equation system consisting of water dynamics equations,silt transport equation,and the equation of bottom topography change were derived.A fully discrete MFE scheme for the discrete_time along characteristics is presented and error estimates are established.The existence and convergence of MFE solution of the discrete current velocity,elevation of the bottom topography,thickness of fluid column,and mass rate of sediment is demonstrated.展开更多
This work presents a locking-free smoothed finite element method(S-FEM)for the simulation of soft matter modelled by the equations of quasi-incompressible hyperelasticity.The proposed method overcomes well-known issue...This work presents a locking-free smoothed finite element method(S-FEM)for the simulation of soft matter modelled by the equations of quasi-incompressible hyperelasticity.The proposed method overcomes well-known issues of standard finite element methods(FEM)in the incompressible limit:the over-estimation of stiffness and sensitivity to severely distorted meshes.The concepts of cell-based,edge-based and node-based S-FEMs are extended in this paper to three-dimensions.Additionally,a cubic bubble function is utilized to improve accuracy and stability.For the bubble function,an additional displacement degree of freedom is added at the centroid of the element.Several numerical studies are performed demonstrating the stability and validity of the proposed approach.The obtained results are compared with standard FEM and with analytical solutions to show the effectiveness of the method.展开更多
The asymptotic convergence of the solution of the parabolic equation is proved. By the eigenvalues estimation, we obtain that the approximate solutions by the finite difference method and the finite element method are...The asymptotic convergence of the solution of the parabolic equation is proved. By the eigenvalues estimation, we obtain that the approximate solutions by the finite difference method and the finite element method are asymptotically convergent. Both methods are considered in continnous time.展开更多
An initial-boundary value problem for shallow equation system consisting of water dynamics equations,silt transport equation, the equation of bottom topography change,and of some boundary and initial conditions is stu...An initial-boundary value problem for shallow equation system consisting of water dynamics equations,silt transport equation, the equation of bottom topography change,and of some boundary and initial conditions is studied, the existence of its generalized solution and semidiscrete mixed finite element(MFE) solution was discussed, and the error estimates of the semidiscrete MFE solution was derived.The error estimates are optimal.展开更多
The vapor deposition chemical reaction processes, which are of extremely extensive applications, can be classified as a mathematical model by the following governing nonlinear partial differential equations containing...The vapor deposition chemical reaction processes, which are of extremely extensive applications, can be classified as a mathematical model by the following governing nonlinear partial differential equations containing velocity vector, temperature field, pressure field, and gas mass field. The mixed finite element (MFE) method is employed to study the system of equations for the vapor deposition chemical reaction processes. The semidiscrete and fully discrete MFE formulations are derived. And the existence and convergence (error estimate) of the semidiscrete and fully discrete MFE solutions are demonstrated. By employing MFE method to treat the system of equations for the vapor deposition chemical reaction processes, the numerical solutions of the velocity vector, the temperature field, the pressure field, and the gas mass field can be found out simultaneously. Thus, these researches are not only of important theoretical means, but also of extremely extensive applied vistas.展开更多
We derived and analyzed a new numerical scheme for the coupled Stokes and Darcy problems by using H(div) conforming elements in the entire domain. The approach employs the mixed finite element method for the Darcy e...We derived and analyzed a new numerical scheme for the coupled Stokes and Darcy problems by using H(div) conforming elements in the entire domain. The approach employs the mixed finite element method for the Darcy equations and a stabilized H(div) finite element method for the Stokes equations. Optimal error estimates for the fluid velocity and pressure are derived. The finite element solutions from the new scheme not only feature a full satisfaction of the continuity equation, which is highly demanded in scientific computing, but also satisfy the mass conservation.展开更多
The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent ...The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.展开更多
文摘This paper focuses on the analytical and numerical asymptotical stability of neutral reaction-diffusion equations with piecewise continuous arguments.First,for the analytical solutions of the equations,we derive their expressions and asymptotical stability criteria.Second,for the semi-discrete and one-parameter fully-discrete finite element methods solving the above equations,we work out the sufficient conditions for assuring that the finite element solutions are asymptotically stable.Finally,with a typical example with numerical experiments,we illustrate the applicability of the obtained theoretical results.
文摘In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.
文摘The transient behavior of a semiconductor device consists of a Poisson equation for the electric potential and of two nonlinear parabolic equations for the electron density and hole density. The electric potential equation is discretized by a mixed finite element method. The electron and hole density equations are treated by implicit-explicit multistep finite element methods. The schemes are very efficient. The optimal order error estimates both in time and space are derived.
文摘In this paper, we represent a new numerical method for solving the nonstationary Stokes equations in an unbounded domain. The technique consists in coupling the boundary integral and finite element methods. The variational formulation and well posedness of the coupling method are obtained. The convergence and optimal estimates for the approximation solution are provided.
基金Project supported by the National Natural Science Foundation of China (No.10471038)
文摘By applying the continuous finite element methods of ordinary differential equations, the linear element methods are proved having second-order pseudo-symplectic scheme and the quadratic element methods are proved having third-order pseudo- symplectic scheme respectively for general Hamiltonian systems, and they both keep energy conservative. The finite element methods are proved to be symplectic as well as energy conservative for linear Hamiltonian systems. The numerical results are in agree-ment with theory.
基金This work was supported in part by the National Science Foundation under grant DMS-1620288。
文摘The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type,coupled through their nonlinear terms.In our previous work[9],we constructed conservative and dissipative finite element methods for these systems and presented a priori error estimates for the semidiscrete schemes.In this sequel,we present a posteriori error estimates for the semidiscrete and fully discrete approximations introduced in[9].The key tool employed to effect our analysis is the dispersive reconstruction devel-oped by Karakashian and Makridakis[20]for related discontinuous Galerkin methods.We conclude by providing a set of numerical experiments designed to validate the a posteriori theory and explore the effectivity of the resulting error indicators.
文摘On the basis of the concept of finite element methods, the rigorous analytical solutions of structural response in terms of the design variables are researched in this paper. The spatial trusses are taken as an example for the solution of the analytical expressions of the explicit displacements which are proved mathematically; then some conclusions are reached that are useful to structural sensitivity analysis and optimization. In the third part of the paper, a generalized geometric programming method is sugguested for the optimal model with the explicit displacement. Finally, the analytical solutions of the displacements of three trusses are given as examples.
基金supported by the National Natural Science Foundation of China (10574156)
文摘Magnetization configurations were calculated under various magnetic fields for nanocrystalline Pr-Fe-B permanent magnets by micromagnetic finite element method.According to the configurations during demagnetization process, the mechanism of magnetization reversal was analyzed.For the Pr2Fe14B with 10 nm grains or its composite with 10vol.% α-Fe, the coercivity was determined by nucleation of reversed domain that took place at grain boundaries.However, for Pr2Fe14B with 30 nm grains, coercivity was controlled by pinning of the nucle-ated domain.For Pr2Fe14B/α-Fe with 30vol.% α-Fe, the demagnetization behavior was characterized by continuous reversal of α-Fe moment.
基金The work is supported by the Guangxi Natural Science Foundation[Grant Numbers 2018GXNSFBA281020,2018GXNSFAA138121]the Doctoral Starting up Foundation of Guilin University of Technology[Grant Number GLUTQD2016044].
文摘In this paper,we consider the numerical schemes for a timefractionalOldroyd-B fluidmodel involving the Caputo derivative.We propose two efficient finite element methods by applying the convolution quadrature in time generated by the backward Euler and the second-order backward difference methods.Error estimates in terms of data regularity are established for both the semidiscrete and fully discrete schemes.Numerical examples for two-dimensional problems further confirmthe robustness of the schemes with first-and second-order accurate in time.
文摘In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.
文摘Chaotic vibrations of flexible non-linear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions(symmetric and non-symmetric)are studied in this work.Reliability of the obtained results is verified by the finite difference method(FDM)and the finite element method(FEM)with the Bubnov-Galerkin approximation for various boundary conditions and various dynamic regimes(regular and non-regular).The influence of boundary conditions on the Euler-Bernoulli beams dynamics is studied mainly,dynamic behavior vs.control parameters { ωp,q0 } is reported,and scenarios of the system transition into chaos are illustrated.
基金supported by NSFC(11571266,91430106,11171168,11071132)NSFC-RGC(China-Hong Kong)(11661161017)
文摘On triangle or quadrilateral meshes, two finite element methods are proposed for solving the Reissner-Mindlin plate problem either by augmenting the Galerkin formulation or modifying the plate-thickness. In these methods, the transverse displacement is approximated by conforming (bi)linear macroelements or (bi)quadratic elements, and the rotation by conforming (bi)linear elements. The shear stress can be locally computed from transverse displacement and rotation. Uniform in plate thickness, optimal error bounds are obtained for the transverse displacement, rotation, and shear stress in their natural norms. Numerical results are presented to illustrate the theoretical results.
基金Subsidized by NSFC (11701343)partially supported by NSFC (11571274,11401466)
文摘In this article,two kinds of expandable parallel finite element methods,based on two-grid discretizations,are given to solve the linear elliptic problems.Compared with the classical local and parallel finite element methods,there are two attractive features of the methods shown in this article:1)a partition of unity is used to generate a series of local and independent subproblems to guarantee the final approximation globally continuous;2)the computational domain of each local subproblem is contained in a ball with radius of O(H)(H is the coarse mesh parameter),which means methods in this article are more suitable for parallel computing in a large parallel computer system.Some a priori error estimation are obtained and optimal error bounds in both H^1-normal and L^2-normal are derived.Finally,numerical results are reported to test and verify the feasibility and validity of our methods.
文摘The mixed finite element(MFE) methods for a shallow water equation system consisting of water dynamics equations,silt transport equation,and the equation of bottom topography change were derived.A fully discrete MFE scheme for the discrete_time along characteristics is presented and error estimates are established.The existence and convergence of MFE solution of the discrete current velocity,elevation of the bottom topography,thickness of fluid column,and mass rate of sediment is demonstrated.
基金Changkye Lee and Jurng-Jae Yee would like to thank the support by Basic Science Research Program through the National Research Foundation(NRF)funded by Korea through Ministry of Education(No.2016R1A6A1A03012812).
文摘This work presents a locking-free smoothed finite element method(S-FEM)for the simulation of soft matter modelled by the equations of quasi-incompressible hyperelasticity.The proposed method overcomes well-known issues of standard finite element methods(FEM)in the incompressible limit:the over-estimation of stiffness and sensitivity to severely distorted meshes.The concepts of cell-based,edge-based and node-based S-FEMs are extended in this paper to three-dimensions.Additionally,a cubic bubble function is utilized to improve accuracy and stability.For the bubble function,an additional displacement degree of freedom is added at the centroid of the element.Several numerical studies are performed demonstrating the stability and validity of the proposed approach.The obtained results are compared with standard FEM and with analytical solutions to show the effectiveness of the method.
文摘The asymptotic convergence of the solution of the parabolic equation is proved. By the eigenvalues estimation, we obtain that the approximate solutions by the finite difference method and the finite element method are asymptotically convergent. Both methods are considered in continnous time.
文摘An initial-boundary value problem for shallow equation system consisting of water dynamics equations,silt transport equation, the equation of bottom topography change,and of some boundary and initial conditions is studied, the existence of its generalized solution and semidiscrete mixed finite element(MFE) solution was discussed, and the error estimates of the semidiscrete MFE solution was derived.The error estimates are optimal.
基金Project supported by the National Natural Science Foundation of China (Nos.10471100 and 40437017)the Science and Technology Foundation of Beijing Jiaotong University
文摘The vapor deposition chemical reaction processes, which are of extremely extensive applications, can be classified as a mathematical model by the following governing nonlinear partial differential equations containing velocity vector, temperature field, pressure field, and gas mass field. The mixed finite element (MFE) method is employed to study the system of equations for the vapor deposition chemical reaction processes. The semidiscrete and fully discrete MFE formulations are derived. And the existence and convergence (error estimate) of the semidiscrete and fully discrete MFE solutions are demonstrated. By employing MFE method to treat the system of equations for the vapor deposition chemical reaction processes, the numerical solutions of the velocity vector, the temperature field, the pressure field, and the gas mass field can be found out simultaneously. Thus, these researches are not only of important theoretical means, but also of extremely extensive applied vistas.
基金The Key Technologies R&D Program ofSichuan Province (No.05GG006-0062)
文摘We derived and analyzed a new numerical scheme for the coupled Stokes and Darcy problems by using H(div) conforming elements in the entire domain. The approach employs the mixed finite element method for the Darcy equations and a stabilized H(div) finite element method for the Stokes equations. Optimal error estimates for the fluid velocity and pressure are derived. The finite element solutions from the new scheme not only feature a full satisfaction of the continuity equation, which is highly demanded in scientific computing, but also satisfy the mass conservation.
文摘The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.