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 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
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
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
The effect of numerical integration in finite element methods applied to a class of nonlinear parabolic equations is considered and some sufficient conditions on the quadrature scheme to ensure that the order of conve...The effect of numerical integration in finite element methods applied to a class of nonlinear parabolic equations is considered and some sufficient conditions on the quadrature scheme to ensure that the order of convergence is unaltered in the presence of numerical integration are given. Optimal L 2 and H 1 estimates for the error and its time derivative are established.展开更多
Error estimates are established for the finite element methods to solve a class of second order nonlinear parabolic equations. Optimal rates of convergence in L 2 and H 1 norms are derived.Meanwhile,the schemes are se...Error estimates are established for the finite element methods to solve a class of second order nonlinear parabolic equations. Optimal rates of convergence in L 2 and H 1 norms are derived.Meanwhile,the schemes are second order correct in time.展开更多
We study the effect of mumerical integration in Finite Element Methods for nonlinear sobolev equations and give some sufficient conditions on the quadrature scheme to ensure that the or- der of convergence is unaltere...We study the effect of mumerical integration in Finite Element Methods for nonlinear sobolev equations and give some sufficient conditions on the quadrature scheme to ensure that the or- der of convergence is unaltered in the presence of mumerical integration.Optimal L2and H1 esti- mates for the error and its time derivative are established.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
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
In this paper,we investigate the error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems.The state and co-state are approximated by the lowest order Raviart-...In this paper,we investigate the error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems.The state and co-state are approximated by the lowest order Raviart-Thomas mixed fi-nite element spaces and the control variable is approximated by piecewise constant functions.We derive L^(2) and L^(∞)-error estimates for the control variable.Moreover,using a recovery operator,we also derive some superconvergence results for the control variable.Finally,a numerical example is given to demonstrate the theoretical results.展开更多
In this paper,we study the mathematical formulation for an optimal control problem governed by a linear parabolic integro-differential equation and present the optimality conditions.We then set up its weak formulation...In this paper,we study the mathematical formulation for an optimal control problem governed by a linear parabolic integro-differential equation and present the optimality conditions.We then set up its weak formulation and the finite element approximation scheme.Based on these we derive the a priori error estimates for its finite element approximation both in H1 and L^(2)norms.Furthermore some numerical tests are presented to verify the theoretical results.展开更多
In this paper,we present an a posteriori error estimates of semilinear quadratic constrained optimal control problems using triangular mixed finite element methods.The state and co-state are approximated by the orde...In this paper,we present an a posteriori error estimates of semilinear quadratic constrained optimal control problems using triangular mixed finite element methods.The state and co-state are approximated by the order k≤1 RaviartThomas mixed finite element spaces and the control is approximated by piecewise constant element.We derive a posteriori error estimates for the coupled state and control approximations.A numerical example is presented in confirmation of the theory.展开更多
The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approx...The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical results.展开更多
This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints.The time ...This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints.The time discretization is based on the backward Euler method.The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions.We derive the superconvergence properties of finite element solutions.By using the superconvergence results,we obtain recovery type a posteriori error estimates.Some numerical examples are presented to verify the theoretical results.展开更多
The purpose of this paper is to investigate the convergence of the mixed finite element method for the initial-boundary value problem for the Sobolev equation Ut-div{aut + b1 u} = f based on the Raviart-Thomas space ...The purpose of this paper is to investigate the convergence of the mixed finite element method for the initial-boundary value problem for the Sobolev equation Ut-div{aut + b1 u} = f based on the Raviart-Thomas space Vh × Wh H(div; × L2(). Optimal order estimates are obtained for the approximation of u, ut, the associated velocity p and divp respectively in L(0,T;L2()), L(0,T;L2()), L(0,T;L2()2), and L2(0, T; L2()). Quasi-optimal order estimates are obtained for the approximations of u, ut in L(0, T; L()) and p in L(0,T; L()2).展开更多
Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/ Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal ...Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/ Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal error order for the porous media flow and a non-optimal error order for the fluid flow. In this article, we give a more rigorous of the error analysis for the fluid flow and obtain the optimal error estimates of the velocity and the pressure.展开更多
In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existenc...In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.展开更多
This paper studies the Galerkin finite element approximations of a class of stochas- tic fractionM differential equations. The discretization in space is done by a standard continuous finite element method and almost ...This paper studies the Galerkin finite element approximations of a class of stochas- tic fractionM differential equations. The discretization in space is done by a standard continuous finite element method and almost optimal order error estimates are obtained. The discretization in time is achieved via the piecewise constant, discontinuous Galerkin method and a Laplace transform convolution quadrature. We give strong convergence error estimates for both semidiscrete and fully discrete schemes. The proof is based on the error estimates for the corresponding deterministic problem. Finally, the numerical example is carried out to verify the theoretical results.展开更多
Error reduction, convergence and optimality are analyzed for adaptive mixed finite element methods (AMFEM) for diffusion equations without marking the oscillation of data. Firstly, the quasi-error, i.e. the sum of t...Error reduction, convergence and optimality are analyzed for adaptive mixed finite element methods (AMFEM) for diffusion equations without marking the oscillation of data. Firstly, the quasi-error, i.e. the sum of the stress variable error and the scaled error estimator, is shown to reduce with a fixed factor between two successive adaptive loops, up to an oscillation. Secondly, the convergence of AMFEM is obtained with respect to the quasi-error plus the divergence of the flux error. Finally, the quasi-optimal convergence rate is established for the total error, i.e. the stress variable error plus the data oscillation.展开更多
In this paper,we consider numerical approximation of a class of nonlinear backward stochastic partial differential equations(BSPDEs).By using finite element methods in the physical space domain and the Euler method in...In this paper,we consider numerical approximation of a class of nonlinear backward stochastic partial differential equations(BSPDEs).By using finite element methods in the physical space domain and the Euler method in the time domain,we propose a spatial finite element semi-discrete scheme and a spatio-temporal full discrete scheme for solving the BSPDEs.Errors of the schemes are rigorously analyzed and theoretical error estimates with convergence rates are obtained.展开更多
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.
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.
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The effect of numerical integration in finite element methods applied to a class of nonlinear parabolic equations is considered and some sufficient conditions on the quadrature scheme to ensure that the order of convergence is unaltered in the presence of numerical integration are given. Optimal L 2 and H 1 estimates for the error and its time derivative are established.
基金National Natural Science Foundation and Doctoral Foundation of Education Ministry of Stat
文摘Error estimates are established for the finite element methods to solve a class of second order nonlinear parabolic equations. Optimal rates of convergence in L 2 and H 1 norms are derived.Meanwhile,the schemes are second order correct in time.
文摘We study the effect of mumerical integration in Finite Element Methods for nonlinear sobolev equations and give some sufficient conditions on the quadrature scheme to ensure that the or- der of convergence is unaltered in the presence of mumerical integration.Optimal L2and H1 esti- mates for the error and its time derivative are established.
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.
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This work is supported by National Science Foundation of China,Foundation for Talent Introduction of Guangdong Provincial University,Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2008)Specialized Research Fund for the Doctoral Program of Higher Education(20114407110009).
文摘In this paper,we investigate the error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems.The state and co-state are approximated by the lowest order Raviart-Thomas mixed fi-nite element spaces and the control variable is approximated by piecewise constant functions.We derive L^(2) and L^(∞)-error estimates for the control variable.Moreover,using a recovery operator,we also derive some superconvergence results for the control variable.Finally,a numerical example is given to demonstrate the theoretical results.
基金W.F.Shen was supported by National Natural Science Foundation of China(Grant:11326226)Nature Science Foundation of Shandong Province(No.ZR2012GM018)D.P.Yang partially was supported by National Natural Science Foundation of China,Grant:11071080.
文摘In this paper,we study the mathematical formulation for an optimal control problem governed by a linear parabolic integro-differential equation and present the optimality conditions.We then set up its weak formulation and the finite element approximation scheme.Based on these we derive the a priori error estimates for its finite element approximation both in H1 and L^(2)norms.Furthermore some numerical tests are presented to verify the theoretical results.
基金supported by Guangdong Provincial‘Zhujiang Scholar Award Project’National Science Foundation of China 10671163+2 种基金the National Basic Research Program under the Grant 2005CB321703Scientific Research Fund of Hunan Provincial Education Department 06A069Hunan Provincial Innovation Foundation for Postgraduate S2008yjscx04。
文摘In this paper,we present an a posteriori error estimates of semilinear quadratic constrained optimal control problems using triangular mixed finite element methods.The state and co-state are approximated by the order k≤1 RaviartThomas mixed finite element spaces and the control is approximated by piecewise constant element.We derive a posteriori error estimates for the coupled state and control approximations.A numerical example is presented in confirmation of the theory.
文摘The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed by quasilinear elliptic partial differential equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical results.
基金supported by Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2008)National Science Foundation of China(10971074)+1 种基金Specialized Research Fund for the Doctoral Program of Higher Education(20114407110009)Hunan Provinical Innovation Foundation for Postgraduate(lx2009 B120)。
文摘This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints.The time discretization is based on the backward Euler method.The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions.We derive the superconvergence properties of finite element solutions.By using the superconvergence results,we obtain recovery type a posteriori error estimates.Some numerical examples are presented to verify the theoretical results.
文摘The purpose of this paper is to investigate the convergence of the mixed finite element method for the initial-boundary value problem for the Sobolev equation Ut-div{aut + b1 u} = f based on the Raviart-Thomas space Vh × Wh H(div; × L2(). Optimal order estimates are obtained for the approximation of u, ut, the associated velocity p and divp respectively in L(0,T;L2()), L(0,T;L2()), L(0,T;L2()2), and L2(0, T; L2()). Quasi-optimal order estimates are obtained for the approximations of u, ut in L(0, T; L()) and p in L(0,T; L()2).
基金Subsidized by NSFC(11571274 and 11171269)the Ph.D.Programs Foundation of Ministry of Education of China(20110201110027)
文摘Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/ Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal error order for the porous media flow and a non-optimal error order for the fluid flow. In this article, we give a more rigorous of the error analysis for the fluid flow and obtain the optimal error estimates of the velocity and the pressure.
基金supported by the National Basic Research Program under the Grant 2005CB321701the National Natural Science Foundation of China under the Grants 60474027 and 10771211.
文摘In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.
文摘This paper studies the Galerkin finite element approximations of a class of stochas- tic fractionM differential equations. The discretization in space is done by a standard continuous finite element method and almost optimal order error estimates are obtained. The discretization in time is achieved via the piecewise constant, discontinuous Galerkin method and a Laplace transform convolution quadrature. We give strong convergence error estimates for both semidiscrete and fully discrete schemes. The proof is based on the error estimates for the corresponding deterministic problem. Finally, the numerical example is carried out to verify the theoretical results.
文摘Error reduction, convergence and optimality are analyzed for adaptive mixed finite element methods (AMFEM) for diffusion equations without marking the oscillation of data. Firstly, the quasi-error, i.e. the sum of the stress variable error and the scaled error estimator, is shown to reduce with a fixed factor between two successive adaptive loops, up to an oscillation. Secondly, the convergence of AMFEM is obtained with respect to the quasi-error plus the divergence of the flux error. Finally, the quasi-optimal convergence rate is established for the total error, i.e. the stress variable error plus the data oscillation.
基金supported by the Science Challenge Project(No.TZ2018001)by National Key R&D Plan(No.2018YFA0703903)by the National Natural Science Foundations of China(under Grants Nos.11901565,11571206,11831010 and 11871068).
文摘In this paper,we consider numerical approximation of a class of nonlinear backward stochastic partial differential equations(BSPDEs).By using finite element methods in the physical space domain and the Euler method in the time domain,we propose a spatial finite element semi-discrete scheme and a spatio-temporal full discrete scheme for solving the BSPDEs.Errors of the schemes are rigorously analyzed and theoretical error estimates with convergence rates are obtained.