In this paper,we consider the initial-boundary value problem(IBVP)for the micropolar Naviers-Stokes equations(MNSE)and analyze a first order fully discrete mixed finite element scheme.We first establish some regularit...In this paper,we consider the initial-boundary value problem(IBVP)for the micropolar Naviers-Stokes equations(MNSE)and analyze a first order fully discrete mixed finite element scheme.We first establish some regularity results for the solution of MNSE,which seem to be not available in the literature.Next,we study a semi-implicit time-discrete scheme for the MNSE and prove L2-H1 error estimates for the time discrete solution.Furthermore,certain regularity results for the time discrete solution are establishes rigorously.Based on these regularity results,we prove the unconditional L2-H1 error estimates for the finite element solution of MNSE.Finally,some numerical examples are carried out to demonstrate both accuracy and efficiency of the fully discrete finite element scheme.展开更多
In this paper,we give improved error estimates for linearized and nonlinear CrankNicolson type finite difference schemes of Ginzburg-Landau equation in two dimensions.For linearized Crank-Nicolson scheme,we use mathem...In this paper,we give improved error estimates for linearized and nonlinear CrankNicolson type finite difference schemes of Ginzburg-Landau equation in two dimensions.For linearized Crank-Nicolson scheme,we use mathematical induction to get unconditional error estimates in discrete L^(2)and H^(1)norm.However,it is not applicable for the nonlinear scheme.Thus,based on a‘cut-off’function and energy analysis method,we get unconditional L^(2)and H^(1)error estimates for the nonlinear scheme,as well as boundedness of numerical solutions.In addition,if the assumption for exact solutions is improved compared to before,unconditional and optimal pointwise error estimates can be obtained by energy analysis method and several Sobolev inequalities.Finally,some numerical examples are given to verify our theoretical analysis.展开更多
In this paper,we consider the Shliomis ferrofluid model and study its numerical approximation.We investigate a first-order energy-stable fully discrete finite element scheme for solving the simplified ferrohydrodynami...In this paper,we consider the Shliomis ferrofluid model and study its numerical approximation.We investigate a first-order energy-stable fully discrete finite element scheme for solving the simplified ferrohydrodynamics(SFHD)equations.First,we establish the well-posedness and some regularity results for the solution of the SFHD model.Next we study the Euler semi-implicit time-discrete scheme for the SFHD systems and derive the L^(2)-H^(1)error estimates for the time-discrete solution.Moreover,certain regularity results for the time-discrete solution are proved rigorously.With the help of these regularity results,we prove the unconditional L^(2)-H^(1)error estimates for the finite element solution of the SFHD model.Finally,some three-dimensional numerical examples are carried out to demonstrate both the accuracy and efficiency of the fully discrete finite element scheme.展开更多
In this paper,we propose a fully discrete finite element method for an incompressible ferrohydrodynamics flow.The constitutive equation we consider,proposed by Rosensweig(2002),models the motion of a magnetic fluid.We...In this paper,we propose a fully discrete finite element method for an incompressible ferrohydrodynamics flow.The constitutive equation we consider,proposed by Rosensweig(2002),models the motion of a magnetic fluid.We develop a semi-implicit,energy-stable scheme to solve this nonlinear system.Using the Leray-Schauder fixed point theorem,we establish the existence and uniqueness of the numerical solutions.Additionally,we prove the unconditional convergence of the numerical scheme through the Aubin-Lions-Simon lemma.Numerical experiments are conducted to verify the convergence of our scheme and to simulate the behavior of ferrohydrodynamic flows.展开更多
This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the n...This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the nonlinear term is mere cubic.The new framework of convergence analysis consists of two steps.In the first step,by truncating the nonlinear term into a global Lipschitz function,an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L2 norm;followed in the second step,the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique,as implies that the two numerical methods are the same one.Under our framework of convergence analysis,with neither any restriction on the grid ratio nor any requirement of the small initial value,we establish the error estimate of the proposed conservative Fourier pseudo-spectral method,while previous work requires the certain restriction for the focusing case.The error bound is proved to be of O(h^(r)+t^(2))with grid size h and time step t.In fact,the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schr¨odinger-type equations.Numerical results are conducted to indicate the accuracy and efficiency of the proposed method,and investigate the effect of the nonlinear term and initial data on the blow-up solution.展开更多
In this paper,we study the finite element approximation for nonlinear thermal equation.Because the nonlinearity of the equation,our theoretical analysis is based on the error of temporal and spatial discretization.We ...In this paper,we study the finite element approximation for nonlinear thermal equation.Because the nonlinearity of the equation,our theoretical analysis is based on the error of temporal and spatial discretization.We consider a fully discrete second order backward difference formula based on a finite element method to approximate the temperature and electric potential,and establish optimal L^(2)error estimates for the fully discrete finite element solution without any restriction on the time-step size.The discrete solution is bounded in infinite norm.Finally,several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.11871467,11471329).
文摘In this paper,we consider the initial-boundary value problem(IBVP)for the micropolar Naviers-Stokes equations(MNSE)and analyze a first order fully discrete mixed finite element scheme.We first establish some regularity results for the solution of MNSE,which seem to be not available in the literature.Next,we study a semi-implicit time-discrete scheme for the MNSE and prove L2-H1 error estimates for the time discrete solution.Furthermore,certain regularity results for the time discrete solution are establishes rigorously.Based on these regularity results,we prove the unconditional L2-H1 error estimates for the finite element solution of MNSE.Finally,some numerical examples are carried out to demonstrate both accuracy and efficiency of the fully discrete finite element scheme.
基金Supported by the National Natural Science Foundation of China(Grant No.11571181)the Research Start-Up Foundation of Nantong University(Grant No.135423602051).
文摘In this paper,we give improved error estimates for linearized and nonlinear CrankNicolson type finite difference schemes of Ginzburg-Landau equation in two dimensions.For linearized Crank-Nicolson scheme,we use mathematical induction to get unconditional error estimates in discrete L^(2)and H^(1)norm.However,it is not applicable for the nonlinear scheme.Thus,based on a‘cut-off’function and energy analysis method,we get unconditional L^(2)and H^(1)error estimates for the nonlinear scheme,as well as boundedness of numerical solutions.In addition,if the assumption for exact solutions is improved compared to before,unconditional and optimal pointwise error estimates can be obtained by energy analysis method and several Sobolev inequalities.Finally,some numerical examples are given to verify our theoretical analysis.
基金supported by the National Natural Science Foundation of China(Nos.12271514,11871467,12161141017)the National Key Research and Development Program of China(2023YFC3705701).
文摘In this paper,we consider the Shliomis ferrofluid model and study its numerical approximation.We investigate a first-order energy-stable fully discrete finite element scheme for solving the simplified ferrohydrodynamics(SFHD)equations.First,we establish the well-posedness and some regularity results for the solution of the SFHD model.Next we study the Euler semi-implicit time-discrete scheme for the SFHD systems and derive the L^(2)-H^(1)error estimates for the time-discrete solution.Moreover,certain regularity results for the time-discrete solution are proved rigorously.With the help of these regularity results,we prove the unconditional L^(2)-H^(1)error estimates for the finite element solution of the SFHD model.Finally,some three-dimensional numerical examples are carried out to demonstrate both the accuracy and efficiency of the fully discrete finite element scheme.
基金supported by National Natural Science Foundation of China(Grant Nos.12071404,11971410,12071402,and 12261131501)Science and Technology Innovation Program of Hunan Province(Grant No.2024RC3158)+3 种基金Young Elite Scientist Sponsorship Program by China Association for Science and Technology(Grant No.2020QNRC001)Key Project of Scientific Research Project of Hunan Provincial Department of Education(Grant No.22A0136)Project of Scientific Research Fund of the Hunan Provincial Science and Technology Department(Grant Nos.2023GK2029 and 2024ZL5017)Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province of China.
文摘In this paper,we propose a fully discrete finite element method for an incompressible ferrohydrodynamics flow.The constitutive equation we consider,proposed by Rosensweig(2002),models the motion of a magnetic fluid.We develop a semi-implicit,energy-stable scheme to solve this nonlinear system.Using the Leray-Schauder fixed point theorem,we establish the existence and uniqueness of the numerical solutions.Additionally,we prove the unconditional convergence of the numerical scheme through the Aubin-Lions-Simon lemma.Numerical experiments are conducted to verify the convergence of our scheme and to simulate the behavior of ferrohydrodynamic flows.
基金Jialing Wang’s work is supported by the National Natural Science Foundation of China(Grant No.11801277)Tingchun Wang’s work is supported by the National Natural Science Foundation of China(Grant No.11571181)+1 种基金the Natural Science Foundation of Jiangsu Province(Grant No.BK20171454)Qing Lan Project.Yushun Wang’s work is supported by the National Natural Science Foundation of China(Grant Nos.11771213 and 12171245).
文摘This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the nonlinear term is mere cubic.The new framework of convergence analysis consists of two steps.In the first step,by truncating the nonlinear term into a global Lipschitz function,an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L2 norm;followed in the second step,the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique,as implies that the two numerical methods are the same one.Under our framework of convergence analysis,with neither any restriction on the grid ratio nor any requirement of the small initial value,we establish the error estimate of the proposed conservative Fourier pseudo-spectral method,while previous work requires the certain restriction for the focusing case.The error bound is proved to be of O(h^(r)+t^(2))with grid size h and time step t.In fact,the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schr¨odinger-type equations.Numerical results are conducted to indicate the accuracy and efficiency of the proposed method,and investigate the effect of the nonlinear term and initial data on the blow-up solution.
文摘In this paper,we study the finite element approximation for nonlinear thermal equation.Because the nonlinearity of the equation,our theoretical analysis is based on the error of temporal and spatial discretization.We consider a fully discrete second order backward difference formula based on a finite element method to approximate the temperature and electric potential,and establish optimal L^(2)error estimates for the fully discrete finite element solution without any restriction on the time-step size.The discrete solution is bounded in infinite norm.Finally,several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.
