The mathematical model of the three-dimensional semiconductor devices of heat conduction is described by a system of four quasilinear partial differential equations for initial boundary value problem. One equation in ...The mathematical model of the three-dimensional semiconductor devices of heat conduction is described by a system of four quasilinear partial differential equations for initial boundary value problem. One equation in elliptic form is for the electric potential; two equations of convection-dominated diffusion type are for the electron and hole concentration; and one heat conduction equation is for temperature. Characteristic finite difference schemes for two kinds of boundary value problems are put forward. By using the thick and thin grids to form a complete set and treating the product threefold-quadratic interpolation, variable time step method with the boundary condition, calculus of variations and the theory of prior estimates and techniques, the optimal error estimates in L2 norm are derived in the approximate solutions.展开更多
The mathematical model of the semiconductor device of heat conduction has been described by a system of four equations. The optimal order estimates in L2 norm are derived for the error in the approximates solution, pu...The mathematical model of the semiconductor device of heat conduction has been described by a system of four equations. The optimal order estimates in L2 norm are derived for the error in the approximates solution, putting fotward a kind of characteristic finite difference fractional step methods.展开更多
In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the un-derlying problem. Unconditional L2-stability...In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the un-derlying problem. Unconditional L2-stability and error estimate of order O？τr＋1＋hk＋1/2？ are obtained when polynomials of degree at most r and k are used for the temporal dis-cretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r＋1 in temporal variable t.展开更多
We develop a family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations,including the characteristic NIPG,OBB,IIPG,and SIPG schemes.The derived schemes possess combined advanta...We develop a family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations,including the characteristic NIPG,OBB,IIPG,and SIPG schemes.The derived schemes possess combined advantages of EulerianLagrangian methods and discontinuous Galerkin methods.An optimal-order error estimate in the L2 norm and a superconvergence estimate in a weighted energy norm are proved for the characteristic NIPG,IIPG,and SIPG scheme.Numerical experiments are presented to confirm the optimal-order spatial and temporal convergence rates of these schemes as proved in the theorems and to show that these schemes compare favorably to the standard NIPG,OBB,IIPG,and SIPG schemes in the context of advection-diffusion equations.展开更多
An implicit discontinuous Galerkin method is introduced to solve the timedomain Maxwell’s equations in metamaterials.The Maxwell’s equations in metamaterials are represented by integral-differential equations.Our sc...An implicit discontinuous Galerkin method is introduced to solve the timedomain Maxwell’s equations in metamaterials.The Maxwell’s equations in metamaterials are represented by integral-differential equations.Our scheme is based on discontinuous Galerkin method in spatial domain and Crank-Nicolson method in temporal domain.The fully discrete numerical scheme is proved to be unconditionally stable.When 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.展开更多
The research of the miscible oil and water displacement problem with moving boundary values is of great value to the history of oil-gas transport and accumulation in the basin evolution as well as to the rational eval...The research of the miscible oil and water displacement problem with moving boundary values is of great value to the history of oil-gas transport and accumulation in the basin evolution as well as to the rational evaluation in prospecting and exploiting oil-gas resources. The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values. For the twodimensional bounded region, the upwind finite difference schemes are proposed. Some techniques, such as the calculus of variations, the change of variables, and the theory of a priori estimates, are used. The optimal orderl2-norm estimates are derived for the errors in the approximate solutions. The research is important both theoretically and practically for the model analysis in the field, the model numerical method, and the software development.展开更多
A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell’s equations.Distinguished from the Runge-Kutta discontinuous Galerkin method(RKDG)and the finite element time domain met...A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell’s equations.Distinguished from the Runge-Kutta discontinuous Galerkin method(RKDG)and the finite element time domain method(FETD),in our scheme,discontinuous Galerkinmethods are used to discretize not only the spatial domain but also the temporal domain.The proposed numerical scheme is proved to be unconditionally stable,and a convergent rate O((△t)^(r+1)+h^(k+1/2))is established under the L^(2)-normwhen polynomials of degree atmost r and k are used for temporal and spatial approximation,respectively.Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction.An ultra-convergence of order(△t)^(2r+1) in time step is observed numerically for the numerical fluxes w.r.t.temporal variable at the grid points.展开更多
In this article,we study a 2D nonlinear time-fractional Rayleigh-Stokes problem,which has an anomalous subdiffusion term,on triangular meshes by quadratic finite volume element schemes.Time-fractional derivative,defin...In this article,we study a 2D nonlinear time-fractional Rayleigh-Stokes problem,which has an anomalous subdiffusion term,on triangular meshes by quadratic finite volume element schemes.Time-fractional derivative,defined by Caputo fractional derivative,is discretized through L2−1σformula,and a two step scheme is used to approximate the time first-order derivative at time tn−α/2,where the nonlinear term is approximated by using a matching linearized difference scheme.A family of quadratic finite volume element schemes with two parameters are proposed for the spatial discretization,where the range of values for two parameters areβ1∈(0,1/2),β2∈(0,2/3).For testing the precision of numerical algorithms,we calculate some numerical examples which have known exact solution or unknown exact solution by several kinds of quadratic finite volume element schemes,and contrast with the results of an existing quadratic finite element scheme by drawing diversified comparison plots and showing the detailed data of L2 error results and convergence orders.Numerical results indicate that,L2 error estimate of one scheme with parameters β_(1)=(3−√3)/6,β2=(6+√3−√21+6√3)/9 is O(h^(3)+△t^(2)),and L^(2) error estimates of other schemes are O(h^(2)+△t^(2)),where h and △t denote the spatial and temporal discretization parameters,respectively.展开更多
We’ll study the FEM for a model for compressible miscible displacement in porous media which includes molecular diffusion and mechanical dispersion in one-dimensional space.A class of vertices-edges-elements interpol...We’ll study the FEM for a model for compressible miscible displacement in porous media which includes molecular diffusion and mechanical dispersion in one-dimensional space.A class of vertices-edges-elements interpolation operator ink is introduced.With the help of ink(not elliptic projection),the optimal error estimate in L∞(J;L2(Ω)) norm of FEM is proved.展开更多
In the paper, we analyze the L2 norm error estimate of lower order finite element methods for the fourth order problem. We prove that the best error estimate in the L2 norm of the finite element solution is of second ...In the paper, we analyze the L2 norm error estimate of lower order finite element methods for the fourth order problem. We prove that the best error estimate in the L2 norm of the finite element solution is of second order, which can not be improved generally. The main ingredients are the saturation condition established for these elements and an identity for the error in the energy norm of the finite element solution. The result holds for most of the popular lower order finite element methods in the literature including： the Powell-Sabin C1 -P2 macro element, the nonconforming Morley element, the C1 -Q2 macro element, the nonconforming rectangle Morley element, and the nonconforming incomplete biquadratic element. In addition, the result actually applies to the nonconforming Adini element, the nonconforming Fraeijs de Veubeke elements, and the nonconforming Wang- Xu element and the Wang-Shi-Xu element provided that the saturation condition holds for them. This result solves one long standing problem in the literature： can the L2 norm error estimate of lower order finite element methods of the fourth order problem be two order higher than the error estimate in the energy norm？展开更多
基金Project supported by the National Scaling Program,the National Eighth-Five Year Tackling Key Problems Program and the Doctoral Found of the National Education Commission.
文摘The mathematical model of the three-dimensional semiconductor devices of heat conduction is described by a system of four quasilinear partial differential equations for initial boundary value problem. One equation in elliptic form is for the electric potential; two equations of convection-dominated diffusion type are for the electron and hole concentration; and one heat conduction equation is for temperature. Characteristic finite difference schemes for two kinds of boundary value problems are put forward. By using the thick and thin grids to form a complete set and treating the product threefold-quadratic interpolation, variable time step method with the boundary condition, calculus of variations and the theory of prior estimates and techniques, the optimal error estimates in L2 norm are derived in the approximate solutions.
文摘The mathematical model of the semiconductor device of heat conduction has been described by a system of four equations. The optimal order estimates in L2 norm are derived for the error in the approximates solution, putting fotward a kind of characteristic finite difference fractional step methods.
基金supported by NSFC(11341002)NSFC(11171104,10871066)+1 种基金the Construct Program of the Key Discipline in Hunansupported in part by US National Science Foundation under Grant DMS-1115530
文摘In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the un-derlying problem. Unconditional L2-stability and error estimate of order O？τr＋1＋hk＋1/2？ are obtained when polynomials of degree at most r and k are used for the temporal dis-cretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r＋1 in temporal variable t.
文摘We develop a family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations,including the characteristic NIPG,OBB,IIPG,and SIPG schemes.The derived schemes possess combined advantages of EulerianLagrangian methods and discontinuous Galerkin methods.An optimal-order error estimate in the L2 norm and a superconvergence estimate in a weighted energy norm are proved for the characteristic NIPG,IIPG,and SIPG scheme.Numerical experiments are presented to confirm the optimal-order spatial and temporal convergence rates of these schemes as proved in the theorems and to show that these schemes compare favorably to the standard NIPG,OBB,IIPG,and SIPG schemes in the context of advection-diffusion equations.
基金supported by the National Natural Science Foundation of China(Grant Nos.11171104,91430107)the Construct Program of the Key Discipline in Hunan.This first author is supported by Hunan Provincial Innovation Foundation for Postgraduate under Grant CX2013B217.
文摘An implicit discontinuous Galerkin method is introduced to solve the timedomain Maxwell’s equations in metamaterials.The Maxwell’s equations in metamaterials are represented by integral-differential equations.Our scheme is based on discontinuous Galerkin method in spatial domain and Crank-Nicolson method in temporal domain.The fully discrete numerical scheme is proved to be unconditionally stable.When 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.
基金supported by the Major State Basic Research Development Program of China(No.G19990328)the National Key Technologies R&D Program of China (No.20050200069)+1 种基金the National Natural Science Foundation of China (Nos.10771124 and 10372052)the Ph. D. Pro-grams Foundation of Ministry of Education of China (No.20030422047)
文摘The research of the miscible oil and water displacement problem with moving boundary values is of great value to the history of oil-gas transport and accumulation in the basin evolution as well as to the rational evaluation in prospecting and exploiting oil-gas resources. The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values. For the twodimensional bounded region, the upwind finite difference schemes are proposed. Some techniques, such as the calculus of variations, the change of variables, and the theory of a priori estimates, are used. The optimal orderl2-norm estimates are derived for the errors in the approximate solutions. The research is important both theoretically and practically for the model analysis in the field, the model numerical method, and the software development.
基金supported by the NSFC(11171104 and 10871066)the Science and Technology Grant of Guizhou Province(LKS[2010]05)+2 种基金supported by the NSFC(11171104 and 10871066)Hunan Provincial Innovation Foundation for Postgraduate(#CX2010B211).supported by the US National Science Foundation through grant DMS-1115530the Ministry of Education of China through the Changjiang Scholars program,the Guangdong Provincial Government of China through the”Computational Science Innovative Research Team”program,and Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University.
文摘A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell’s equations.Distinguished from the Runge-Kutta discontinuous Galerkin method(RKDG)and the finite element time domain method(FETD),in our scheme,discontinuous Galerkinmethods are used to discretize not only the spatial domain but also the temporal domain.The proposed numerical scheme is proved to be unconditionally stable,and a convergent rate O((△t)^(r+1)+h^(k+1/2))is established under the L^(2)-normwhen polynomials of degree atmost r and k are used for temporal and spatial approximation,respectively.Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction.An ultra-convergence of order(△t)^(2r+1) in time step is observed numerically for the numerical fluxes w.r.t.temporal variable at the grid points.
基金This work was partially supported by the National Natural Science Foundation of China(No.11871009).
文摘In this article,we study a 2D nonlinear time-fractional Rayleigh-Stokes problem,which has an anomalous subdiffusion term,on triangular meshes by quadratic finite volume element schemes.Time-fractional derivative,defined by Caputo fractional derivative,is discretized through L2−1σformula,and a two step scheme is used to approximate the time first-order derivative at time tn−α/2,where the nonlinear term is approximated by using a matching linearized difference scheme.A family of quadratic finite volume element schemes with two parameters are proposed for the spatial discretization,where the range of values for two parameters areβ1∈(0,1/2),β2∈(0,2/3).For testing the precision of numerical algorithms,we calculate some numerical examples which have known exact solution or unknown exact solution by several kinds of quadratic finite volume element schemes,and contrast with the results of an existing quadratic finite element scheme by drawing diversified comparison plots and showing the detailed data of L2 error results and convergence orders.Numerical results indicate that,L2 error estimate of one scheme with parameters β_(1)=(3−√3)/6,β2=(6+√3−√21+6√3)/9 is O(h^(3)+△t^(2)),and L^(2) error estimates of other schemes are O(h^(2)+△t^(2)),where h and △t denote the spatial and temporal discretization parameters,respectively.
基金This research is supported by the Foundation for Talents for Next Century of Shandong University
文摘We’ll study the FEM for a model for compressible miscible displacement in porous media which includes molecular diffusion and mechanical dispersion in one-dimensional space.A class of vertices-edges-elements interpolation operator ink is introduced.With the help of ink(not elliptic projection),the optimal error estimate in L∞(J;L2(Ω)) norm of FEM is proved.
文摘In the paper, we analyze the L2 norm error estimate of lower order finite element methods for the fourth order problem. We prove that the best error estimate in the L2 norm of the finite element solution is of second order, which can not be improved generally. The main ingredients are the saturation condition established for these elements and an identity for the error in the energy norm of the finite element solution. The result holds for most of the popular lower order finite element methods in the literature including： the Powell-Sabin C1 -P2 macro element, the nonconforming Morley element, the C1 -Q2 macro element, the nonconforming rectangle Morley element, and the nonconforming incomplete biquadratic element. In addition, the result actually applies to the nonconforming Adini element, the nonconforming Fraeijs de Veubeke elements, and the nonconforming Wang- Xu element and the Wang-Shi-Xu element provided that the saturation condition holds for them. This result solves one long standing problem in the literature： can the L2 norm error estimate of lower order finite element methods of the fourth order problem be two order higher than the error estimate in the energy norm？
