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
The fractional optimal control problem leads to significantly increased computational complexity compared to the corresponding classical integer-order optimal control problem,due to the global properties of fractional...The fractional optimal control problem leads to significantly increased computational complexity compared to the corresponding classical integer-order optimal control problem,due to the global properties of fractional differential operators.In this paper,we focus on an optimal control problem governed by fractional differential equations with an integral constraint on the state variable.By the proposed first-order optimality condition consisting of a Lagrange multiplier,we design a spectral Galerkin discrete scheme with weighted orthogonal Jacobi polynomials to approximate the resulting state and adjoint state equations.Furthermore,a priori error estimates for state,adjoint state and control variables are discussed in details.Illustrative numerical tests are given to demonstrate the validity and applicability of our proposed approximations and theoretical results.展开更多
In this paper, a constrained distributed optimal control problem governed by a first- order elliptic system is considered. Least-squares mixed finite element methods, which are not subject to the Ladyzhenkaya-Babuska-...In this paper, a constrained distributed optimal control problem governed by a first- order elliptic system is considered. Least-squares mixed finite element methods, which are not subject to the Ladyzhenkaya-Babuska-Brezzi consistency condition, are used for solving the elliptic system with two unknown state variables. By adopting the Lagrange multiplier approach, continuous and discrete optimality systems including a primal state equation, an adjoint state equation, and a variational inequality for the optimal control are derived, respectively. Both the discrete state equation and discrete adjoint state equation yield a symmetric and positive definite linear algebraic system. Thus, the popular solvers such as preconditioned conjugate gradient （PCG） and algebraic multi-grid （AMG） can be used for rapid solution. Optimal a priori error estimates are obtained, respectively, for the control function in L2 （Ω）-norm, for the original state and adjoint state in H1 （Ω）-norm, and for the flux state and adjoint flux state in H（div; Ω）-norm. Finally, we use one numerical example to validate the theoretical findings.展开更多
In this paper we deal with the convergence analysis of the finite element method for an elliptic penalized unilateral obstacle optimal control problem where the control and the obstacle coincide.Error estimates are es...In this paper we deal with the convergence analysis of the finite element method for an elliptic penalized unilateral obstacle optimal control problem where the control and the obstacle coincide.Error estimates are established for both state and control variables.We apply a fixed point type iteration method to solve the discretized problem.To corroborate our error estimations and the eficiency of our algorithms,the convergence results and numerical experiments are illustrated by concrete examples.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
The subject of this work is to propose adaptive finite element methods based on an optimal maximum norm error control estimate.Using estimators of the local regularity of the unknown exact solution derived from comput...The subject of this work is to propose adaptive finite element methods based on an optimal maximum norm error control estimate.Using estimators of the local regularity of the unknown exact solution derived from computed approximate solutions,the proposed procedures are analyzed in detail for a non-trivial class of corner problems and shown to be efficient in the sense that they generate the correct type of refinement and lead to the desired control under consideration.展开更多
In this paper, we propose the nonconforming virtual element method (NCVEM) discretization for the pointwise control constraint optimal control problem governed by elliptic equations. Based on the NCVEM approximation o...In this paper, we propose the nonconforming virtual element method (NCVEM) discretization for the pointwise control constraint optimal control problem governed by elliptic equations. Based on the NCVEM approximation of state equation and the variational discretization of control variables, we construct a virtual element discrete scheme. For the state, adjoint state and control variable, we obtain the corresponding prior estimate in H<sup>1</sup> and L<sup>2</sup> norms. Finally, some numerical experiments are carried out to support the theoretical results.展开更多
Finite Element Method (FEM), when applied to solve problems, has faced some challenges over the years, such as time consumption and the complexity of assumptions. In particular, the making of assumptions has had a sig...Finite Element Method (FEM), when applied to solve problems, has faced some challenges over the years, such as time consumption and the complexity of assumptions. In particular, the making of assumptions has had a significant influence on the accuracy of the method, making it mandatory to carry out sensitivity analysis. The sensitivity analysis helps to identify the level of impact the assumptions have on the method. However, sensitivity analysis via FEM can be very challenging. A priori error estimation, an integral part of FEM, is a basic mathematical tool for predicting the accuracy of numerical solutions. By understanding the relationship between the mesh size, the order of basis functions, and the resulting error, practitioners can effectively design and apply FEM to solve complex Partial Differential Equations (PDEs) with confidence in the reliability of their results. Thus, the coercive property and A priori error estimation based on the L1 formula on a mesh in time and the Mamadu-Njoseh basis functions in space are investigated for a linearly distributed time-order fractional telegraph equation with restricted initial conditions. For this purpose, we constructed a mathematical proof of the coercive property for the fully discretized scheme. Also, we stated and proved a cardinal theorem for a priori error estimation of the approximate solution for the fully discretized scheme. We noticed the role of the restricted initial conditions imposed on the solution in the analysis of a priori error estimation.展开更多
In this paper, we study a weakly over-penalized interior penalty method for non-self- adjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a resi...In this paper, we study a weakly over-penalized interior penalty method for non-self- adjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a residual-based a posteriori error estimator, which is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to validate our theoretical analysis.展开更多
In this paper,we discuss the conforming finite element method for a modified interior transmission eigenvalues problem.We present a complete theoretical analysis for the method,including the a priori and a posteriori ...In this paper,we discuss the conforming finite element method for a modified interior transmission eigenvalues problem.We present a complete theoretical analysis for the method,including the a priori and a posteriori error estimates.The theoretical analysis is conducted under the assumption of low regularity on the solution.We prove the reliability and efficiency of the a posteriori error estimators for eigenfunctions up to higher order terms,and we also analyze the reliability of estimators for eigenvalues.Finally,we report numerical experiments to show that our posteriori error estimator is effective and the approximations can reach the optimal convergence order.The numerical results also indicate that the conforming finite element eigenvalues approximate the exact ones from below,and there exists a monotonic relationship between the conforming finite element eigenvalues and the refractive index through numerical experiments.展开更多
In this paper,we propose and analyze the interior penalty discontinuous Galerkin method for H(div)-elliptic problem.An optimal a priori error estimate in the energy norm is proved.In addition,a residual-based a poster...In this paper,we propose and analyze the interior penalty discontinuous Galerkin method for H(div)-elliptic problem.An optimal a priori error estimate in the energy norm is proved.In addition,a residual-based a posteriori error estimator is obtained.The estimator is proved to be both reliable and efficient in the energy norm.Some numerical testes are presented to demonstrate the effectiveness of our method.展开更多
In this paper, we discuss virtual element method (VEM) approximation of optimal control problem governed by Brinkman equations with control constraints. Based on the polynomial projections and variational discretizati...In this paper, we discuss virtual element method (VEM) approximation of optimal control problem governed by Brinkman equations with control constraints. Based on the polynomial projections and variational discretization of the control variable, we build up the virtual element discrete scheme of the optimal control problem and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L<sup>2</sup> and H<sup>1</sup> norm are derived. The theoretical findings are illustrated by the numerical experiments.展开更多
This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin(LDG)method for two-dimensional semilinear second-order elliptic problems of the form−Δu=f(x,y,u)on Cartesia...This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin(LDG)method for two-dimensional semilinear second-order elliptic problems of the form−Δu=f(x,y,u)on Cartesian grids.By introducing special GaussRadau projections and using duality arguments,we obtain,under some suitable choice of numerical fuxes,the optimal convergence order in L2-norm of O(h^(p+1))for the LDG solution and its gradient,when tensor product polynomials of degree at most p and grid size h are used.Moreover,we prove that the LDG solutions are superconvergent with an order p+2 toward particular Gauss-Radau projections of the exact solutions.Finally,we show that the error between the gradient of the LDG solution and the gradient of a special Gauss-Radau projection of the exact solution achieves(p+1)-th order superconvergence.Some numerical experiments are performed to illustrate the theoretical results.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
Nearly all inf-sup stable mixed finite elements for the incompressible Stokes equations relax the divergence constraint. The price to pay is that a priori estimates for the ve- locity error become pressure-dependent, ...Nearly all inf-sup stable mixed finite elements for the incompressible Stokes equations relax the divergence constraint. The price to pay is that a priori estimates for the ve- locity error become pressure-dependent, while divergence-free mixed finite elements de- liver pressure-independent estimates. A recently introduced new variational crime using lowest-order Raviart-Thomas velocity reconstructions delivers a much more robust modi- fied Crouzeix-Raviart element, obeying an optimal pressure-independent discrete H1 ve- locity estimate. Refining this approach, a more sophisticated variational crime employing the lowest-order BDM element is proposed, which also allows proving an optimal pressure- independent L2 velocity error. Numerical examples confirm the analysis and demonstrate the improved robustness in the Navier-Stokes case.展开更多
Numerical atomic orbitals have been successfully used in molecular simulations as a basis set,which provides a nature,physical description of the electronic states and is suitable for ■(N)calculations based on the st...Numerical atomic orbitals have been successfully used in molecular simulations as a basis set,which provides a nature,physical description of the electronic states and is suitable for ■(N)calculations based on the strictly localized property.This paper presents a numerical analysis for some simplified atomic orbitals,with polynomial-type and confined Hydrogen-like radial basis functions respectively.We give some a priori error estimates to understand why numerical atomic orbitals are computationally efficient in electronic structure calculations.展开更多
A posteriori error computations in the space-time coupled and space-time decoupled finite element methods for initial value problems are essential:1)to determine the accuracy of the computed evolution,2)if the errors ...A posteriori error computations in the space-time coupled and space-time decoupled finite element methods for initial value problems are essential:1)to determine the accuracy of the computed evolution,2)if the errors in the coupled solutions are higher than an acceptable threshold,then a posteriori error computations provide measures for designing adaptive processes to improve the accuracy of the solution.How well the space-time approximation in each of the two methods satisfies the equations in the mathematical model over the space-time domain in the point wise sense is the absolute measure of the accuracy of the computed solution.When L2-norm of the space-time residual over the space-time domain of the computations approaches zero,the approximation φh(x,t)(,)→φ(x,t),the theoretical solution.Thus,the proximity of ||E||L_(2) ,the L_(2)-norm of the space-time residual function,to zero is a measure of the accuracy or the error in the computed solution.In this paper,we present a methodology and a computational framework for computing L2 E in the a posteriori error computations for both space-time coupled and space-time decoupled finite element methods.It is shown that the proposed a posteriori computations require h,p,k framework in both space-time coupled as well as space-time decoupled finite element methods to ensure that space-time integrals over space-time discretization are Riemann,hence the proposed a posteriori computations can not be performed in finite difference and finite volume methods of solving initial value problems.High-order global differentiability in time in the integration methods is essential in space-time decoupled method for posterior computations.This restricts the use of methods like Euler’s method,Runge-Kutta methods,etc.,in the time integration of ODE’s in time.Mathematical and computational details including model problem studies are presented in the paper.To authors knowledge,it is the first presentation of the proposed a posteriori error computation methodology and computational infrastructure for initial value problems.展开更多
This paper is concerned with an ill-posed problem which results from the area of molecular imaging and is known as BLT problem. Using Tikhonov regularization technique, a quadratic optimization problem can be formulat...This paper is concerned with an ill-posed problem which results from the area of molecular imaging and is known as BLT problem. Using Tikhonov regularization technique, a quadratic optimization problem can be formulated. We provide an improved error estimate for the finite element approximation of the regularized optimization problem. Some numerical examples are presented to demonstrate our theoretical results.展开更多
This paper aims to construct a two-grid scheme of fully discretized expanded mixed finite element methods for optimal control problems governed by parabolic integro-differential equations and discuss a priori error es...This paper aims to construct a two-grid scheme of fully discretized expanded mixed finite element methods for optimal control problems governed by parabolic integro-differential equations and discuss a priori error estimates.The state variables and co-state variables are discretized by the lowest order Raviart-Thomas mixed finite element,and the control variable is approximated by piecewise constant functions.The time derivative is discretized by the backward Euler method.Firstly,we define some new mixed elliptic projections and prove the corresponding error estimates which play an important role in subsequent convergence analysis.Secondly,we derive a priori error estimates for all variables.Thirdly,we present a two-grid scheme and analyze its convergence.In the two-grid scheme,the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid and the solution of a decoupled linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.At last,a numerical example is presented to verify the theoretical results.展开更多
In this paper,we investigate a meshless approximation,the interpolating element-free Galerkin method,for an optimal control problem governed by fourth-order parabolic partial differential equations.The state,co-state ...In this paper,we investigate a meshless approximation,the interpolating element-free Galerkin method,for an optimal control problem governed by fourth-order parabolic partial differential equations.The state,co-state and control variables are spatially discretized by an improved moving least squares approximation that satisfies the interpolation property,and time is discretized by a backward-Euler method.We derive some a priori error estimates for both the control and state approximations.Numerical experiments are presented to verify the theoretical results.展开更多
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.
基金This work was partly supported by National Natural Science Foundation of China(Grant Nos.:12101283,12271233 and 12171287)Natural Science Foundation of Shandong Province(Grant Nos.:ZR2019YQ05,2019KJI003,and ZR2016JL004).
文摘The fractional optimal control problem leads to significantly increased computational complexity compared to the corresponding classical integer-order optimal control problem,due to the global properties of fractional differential operators.In this paper,we focus on an optimal control problem governed by fractional differential equations with an integral constraint on the state variable.By the proposed first-order optimality condition consisting of a Lagrange multiplier,we design a spectral Galerkin discrete scheme with weighted orthogonal Jacobi polynomials to approximate the resulting state and adjoint state equations.Furthermore,a priori error estimates for state,adjoint state and control variables are discussed in details.Illustrative numerical tests are given to demonstrate the validity and applicability of our proposed approximations and theoretical results.
基金Acknowledgements. The authors would like to thank the editor and the anonymous referee for their valuable comments and suggestions on an earlier version of this paper. The work of Hongxing Rui （corresponding author） was supported by the NationM Natural Science Founda- tion of China （No. 11171190）. The work of Hongfei Fu was supported by the National Natural Science Foundation of China （No. 11201485）, the Promotive Research Fund for Excellent Young and Middle-aged Scientists of Shandong Province （No. BS2013NJ001）, and the Fundamental Research Funds for the Central Universities （No. 14CX02217A）.
文摘In this paper, a constrained distributed optimal control problem governed by a first- order elliptic system is considered. Least-squares mixed finite element methods, which are not subject to the Ladyzhenkaya-Babuska-Brezzi consistency condition, are used for solving the elliptic system with two unknown state variables. By adopting the Lagrange multiplier approach, continuous and discrete optimality systems including a primal state equation, an adjoint state equation, and a variational inequality for the optimal control are derived, respectively. Both the discrete state equation and discrete adjoint state equation yield a symmetric and positive definite linear algebraic system. Thus, the popular solvers such as preconditioned conjugate gradient （PCG） and algebraic multi-grid （AMG） can be used for rapid solution. Optimal a priori error estimates are obtained, respectively, for the control function in L2 （Ω）-norm, for the original state and adjoint state in H1 （Ω）-norm, and for the flux state and adjoint flux state in H（div; Ω）-norm. Finally, we use one numerical example to validate the theoretical findings.
文摘In this paper we deal with the convergence analysis of the finite element method for an elliptic penalized unilateral obstacle optimal control problem where the control and the obstacle coincide.Error estimates are established for both state and control variables.We apply a fixed point type iteration method to solve the discretized problem.To corroborate our error estimations and the eficiency of our algorithms,the convergence results and numerical experiments are illustrated by concrete examples.
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.
文摘The subject of this work is to propose adaptive finite element methods based on an optimal maximum norm error control estimate.Using estimators of the local regularity of the unknown exact solution derived from computed approximate solutions,the proposed procedures are analyzed in detail for a non-trivial class of corner problems and shown to be efficient in the sense that they generate the correct type of refinement and lead to the desired control under consideration.
文摘In this paper, we propose the nonconforming virtual element method (NCVEM) discretization for the pointwise control constraint optimal control problem governed by elliptic equations. Based on the NCVEM approximation of state equation and the variational discretization of control variables, we construct a virtual element discrete scheme. For the state, adjoint state and control variable, we obtain the corresponding prior estimate in H<sup>1</sup> and L<sup>2</sup> norms. Finally, some numerical experiments are carried out to support the theoretical results.
文摘Finite Element Method (FEM), when applied to solve problems, has faced some challenges over the years, such as time consumption and the complexity of assumptions. In particular, the making of assumptions has had a significant influence on the accuracy of the method, making it mandatory to carry out sensitivity analysis. The sensitivity analysis helps to identify the level of impact the assumptions have on the method. However, sensitivity analysis via FEM can be very challenging. A priori error estimation, an integral part of FEM, is a basic mathematical tool for predicting the accuracy of numerical solutions. By understanding the relationship between the mesh size, the order of basis functions, and the resulting error, practitioners can effectively design and apply FEM to solve complex Partial Differential Equations (PDEs) with confidence in the reliability of their results. Thus, the coercive property and A priori error estimation based on the L1 formula on a mesh in time and the Mamadu-Njoseh basis functions in space are investigated for a linearly distributed time-order fractional telegraph equation with restricted initial conditions. For this purpose, we constructed a mathematical proof of the coercive property for the fully discretized scheme. Also, we stated and proved a cardinal theorem for a priori error estimation of the approximate solution for the fully discretized scheme. We noticed the role of the restricted initial conditions imposed on the solution in the analysis of a priori error estimation.
基金We thank the anonymous referees for their valuable comments and suggestions which lead to an improved presentation of this paper. This work was supported by NSFC under the grant 11371199, 11226334 and 11301275, the Jiangsu Provincial 2011 Program （Collaborative Innovation Center of Climate Change）, the Program of Natural Science Research of Jiangsu Higher Education Institutions of China （Grant No. 12KJB110013）, Natural Science Foundation of Guangdong Province of China （Grant No. S2012040007993） and Educational Commission of Guangdong Province of China （Grant No. 2012LYM0122）.
文摘In this paper, we study a weakly over-penalized interior penalty method for non-self- adjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a residual-based a posteriori error estimator, which is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to validate our theoretical analysis.
基金supported by the National Natural Science Foundation of China(Nos.12261024,11561014)Science and Technology Planning Project of Guizhou Province(Guizhou Kehe fundamental research-ZK[2022]No.324).
文摘In this paper,we discuss the conforming finite element method for a modified interior transmission eigenvalues problem.We present a complete theoretical analysis for the method,including the a priori and a posteriori error estimates.The theoretical analysis is conducted under the assumption of low regularity on the solution.We prove the reliability and efficiency of the a posteriori error estimators for eigenfunctions up to higher order terms,and we also analyze the reliability of estimators for eigenvalues.Finally,we report numerical experiments to show that our posteriori error estimator is effective and the approximations can reach the optimal convergence order.The numerical results also indicate that the conforming finite element eigenvalues approximate the exact ones from below,and there exists a monotonic relationship between the conforming finite element eigenvalues and the refractive index through numerical experiments.
文摘In this paper,we propose and analyze the interior penalty discontinuous Galerkin method for H(div)-elliptic problem.An optimal a priori error estimate in the energy norm is proved.In addition,a residual-based a posteriori error estimator is obtained.The estimator is proved to be both reliable and efficient in the energy norm.Some numerical testes are presented to demonstrate the effectiveness of our method.
文摘In this paper, we discuss virtual element method (VEM) approximation of optimal control problem governed by Brinkman equations with control constraints. Based on the polynomial projections and variational discretization of the control variable, we build up the virtual element discrete scheme of the optimal control problem and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L<sup>2</sup> and H<sup>1</sup> norm are derived. The theoretical findings are illustrated by the numerical experiments.
基金This research was supported by the NASA Nebraska Space Grant(Federal Grant/Award Number 80NSSC20M0112).
文摘This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin(LDG)method for two-dimensional semilinear second-order elliptic problems of the form−Δu=f(x,y,u)on Cartesian grids.By introducing special GaussRadau projections and using duality arguments,we obtain,under some suitable choice of numerical fuxes,the optimal convergence order in L2-norm of O(h^(p+1))for the LDG solution and its gradient,when tensor product polynomials of degree at most p and grid size h are used.Moreover,we prove that the LDG solutions are superconvergent with an order p+2 toward particular Gauss-Radau projections of the exact solutions.Finally,we show that the error between the gradient of the LDG solution and the gradient of a special Gauss-Radau projection of the exact solution achieves(p+1)-th order superconvergence.Some numerical experiments are performed to illustrate the theoretical results.
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.
文摘Nearly all inf-sup stable mixed finite elements for the incompressible Stokes equations relax the divergence constraint. The price to pay is that a priori estimates for the ve- locity error become pressure-dependent, while divergence-free mixed finite elements de- liver pressure-independent estimates. A recently introduced new variational crime using lowest-order Raviart-Thomas velocity reconstructions delivers a much more robust modi- fied Crouzeix-Raviart element, obeying an optimal pressure-independent discrete H1 ve- locity estimate. Refining this approach, a more sophisticated variational crime employing the lowest-order BDM element is proposed, which also allows proving an optimal pressure- independent L2 velocity error. Numerical examples confirm the analysis and demonstrate the improved robustness in the Navier-Stokes case.
基金The research for this paper has been enabled by the Alexander von Humboldt Foundation,whose support for the long term visit of Huajie Chen at Technische Universit¨at Berlin is gratefully acknowledged.
文摘Numerical atomic orbitals have been successfully used in molecular simulations as a basis set,which provides a nature,physical description of the electronic states and is suitable for ■(N)calculations based on the strictly localized property.This paper presents a numerical analysis for some simplified atomic orbitals,with polynomial-type and confined Hydrogen-like radial basis functions respectively.We give some a priori error estimates to understand why numerical atomic orbitals are computationally efficient in electronic structure calculations.
基金grateful for the facilities provided by the Computational Mechanics Laboratory of the Department of Mechanical Engineering.
文摘A posteriori error computations in the space-time coupled and space-time decoupled finite element methods for initial value problems are essential:1)to determine the accuracy of the computed evolution,2)if the errors in the coupled solutions are higher than an acceptable threshold,then a posteriori error computations provide measures for designing adaptive processes to improve the accuracy of the solution.How well the space-time approximation in each of the two methods satisfies the equations in the mathematical model over the space-time domain in the point wise sense is the absolute measure of the accuracy of the computed solution.When L2-norm of the space-time residual over the space-time domain of the computations approaches zero,the approximation φh(x,t)(,)→φ(x,t),the theoretical solution.Thus,the proximity of ||E||L_(2) ,the L_(2)-norm of the space-time residual function,to zero is a measure of the accuracy or the error in the computed solution.In this paper,we present a methodology and a computational framework for computing L2 E in the a posteriori error computations for both space-time coupled and space-time decoupled finite element methods.It is shown that the proposed a posteriori computations require h,p,k framework in both space-time coupled as well as space-time decoupled finite element methods to ensure that space-time integrals over space-time discretization are Riemann,hence the proposed a posteriori computations can not be performed in finite difference and finite volume methods of solving initial value problems.High-order global differentiability in time in the integration methods is essential in space-time decoupled method for posterior computations.This restricts the use of methods like Euler’s method,Runge-Kutta methods,etc.,in the time integration of ODE’s in time.Mathematical and computational details including model problem studies are presented in the paper.To authors knowledge,it is the first presentation of the proposed a posteriori error computation methodology and computational infrastructure for initial value problems.
基金the National Natural Science Foundation of China (No. 60474027,10301003 and 10771211)the National Basic Research Program under the Grant 2005CB321701
文摘This paper is concerned with an ill-posed problem which results from the area of molecular imaging and is known as BLT problem. Using Tikhonov regularization technique, a quadratic optimization problem can be formulated. We provide an improved error estimate for the finite element approximation of the regularized optimization problem. Some numerical examples are presented to demonstrate our theoretical results.
基金supported by National Natural Science Foundation of China(No.12571388)the Visiting scholar program of National Natural Science Foundation of China(No.12426616)Natural Science Research Start-up Foundation of Recruiting Talents of Nanjing University of Posts and Telecommunications(No.NY223127)。
文摘This paper aims to construct a two-grid scheme of fully discretized expanded mixed finite element methods for optimal control problems governed by parabolic integro-differential equations and discuss a priori error estimates.The state variables and co-state variables are discretized by the lowest order Raviart-Thomas mixed finite element,and the control variable is approximated by piecewise constant functions.The time derivative is discretized by the backward Euler method.Firstly,we define some new mixed elliptic projections and prove the corresponding error estimates which play an important role in subsequent convergence analysis.Secondly,we derive a priori error estimates for all variables.Thirdly,we present a two-grid scheme and analyze its convergence.In the two-grid scheme,the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid and the solution of a decoupled linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.At last,a numerical example is presented to verify the theoretical results.
基金supported by the National Natural Science Foundation of China(Grant 11871312)the Natural Science Foundation of Shandong Province,China(Grant ZR2023MA086).
文摘In this paper,we investigate a meshless approximation,the interpolating element-free Galerkin method,for an optimal control problem governed by fourth-order parabolic partial differential equations.The state,co-state and control variables are spatially discretized by an improved moving least squares approximation that satisfies the interpolation property,and time is discretized by a backward-Euler method.We derive some a priori error estimates for both the control and state approximations.Numerical experiments are presented to verify the theoretical results.
