In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and ...In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and co-state variables, and piecewise constant function is used to approximate control variables. Generally, the optimal conditions for the problem are solved iteratively until the control variable reaches error tolerance. In order to calculate all the variables individually and parallelly, we introduce a gradient recovery based two-grid method. First, we solve the small scaled optimal control problem on coarse grids. Next, we use the gradient recovery technique to recover the gradients of state and co-state variables. Finally, using the recovered variables, we solve the large scaled optimal control problem for all variables independently. Moreover, we estimate priori error for the proposed scheme, and use an example to validate the theoretical results.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
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
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
In the optimal control problem of nonlinear dynamical system,the Hamiltonian formulation is useful and powerful to solve an optimal control force.However,the resulting Euler-Lagrange equations are not easy to solve,wh...In the optimal control problem of nonlinear dynamical system,the Hamiltonian formulation is useful and powerful to solve an optimal control force.However,the resulting Euler-Lagrange equations are not easy to solve,when the performance index is complicated,because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations.To be a numerical method,it is hard to exactly preserve all the specified conditions,which might deteriorate the accuracy of numerical solution.With this in mind,we develop a novel algorithm to find the solution of the optimal control problem of nonlinear Duffing oscillator,which can exactly satisfy all the required conditions for the minimality of the performance index.A new idea of shape functions method(SFM)is introduced,from which we can transform the optimal control problems to the initial value problems for the new variables,whose initial values are given arbitrarily,and meanwhile the terminal values are determined iteratively.Numerical examples confirm the high-performance of the iterative algorithms based on the SFM,which are convergence fast,and also provide very accurate solutions.The new algorithm is robust,even large noise is imposed on the input data.展开更多
In this paper, we consider a fully discrete finite element approximation for time fractional optimal control problems. The state and adjoint state are approximated by triangular linear fi nite elements in space and &l...In this paper, we consider a fully discrete finite element approximation for time fractional optimal control problems. The state and adjoint state are approximated by triangular linear fi nite elements in space and <em>L</em>1 scheme in time. The control is obtained by the variational discretization technique. The main purpose of this work is to derive the convergence and superconvergence. A numerical example is presented to validate our theoretical results.展开更多
This paper deals with the optimal control problems of systems governed by a parabolic variational inequality coupled with a semilinear parabolic differential equations. The maximum principle and some kind of approxima...This paper deals with the optimal control problems of systems governed by a parabolic variational inequality coupled with a semilinear parabolic differential equations. The maximum principle and some kind of approximate controllability are studied.展开更多
A kind of direct methods is presented for the solution of optimal control problems with state constraints. These methods are sequential quadratic programming methods. At every iteration a quadratic programming which i...A kind of direct methods is presented for the solution of optimal control problems with state constraints. These methods are sequential quadratic programming methods. At every iteration a quadratic programming which is obtained by quadratic approximation to Lagrangian function and linear approximations to constraints is solved to get a search direction for a merit function. The merit function is formulated by augmenting the Lagrangian function with a penalty term. A line search is carried out along the search direction to determine a step length such that the merit function is decreased. The methods presented in this paper include continuous sequential quadratic programming methods and discreate sequential quadratic programming methods.展开更多
In this paper,we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random coefficients.There are two contributions of thi...In this paper,we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random coefficients.There are two contributions of this paper.Firstly,we establish a scheme to approximate the optimality system by using the finite volume element method in the physical space and the meshfree method in the probability space,which is competitive for high-dimensional random inputs.Secondly,the a priori error estimates are derived for the state,the co-state and the control variables.Some numerical tests are carried out to confirm the theoretical results and demonstrate the efficiency of the proposed method.展开更多
This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain.Such problems usually possess low regularity in the state variable due to t...This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain.Such problems usually possess low regularity in the state variable due to the presence of measure data and the nonconvex nature of the domain.The low regularity of the solution allows the finite element approximations to converge at lower orders.We prove the existence,uniqueness and regularity results for the solution to the control problem satisfying the first order optimality condition.For our error analysis we have used piecewise linear elements for the approximation of the state and co-state variables,whereas piecewise constant functions are employed to approximate the control variable.The temporal discretization is based on the implicit Euler scheme.We derive both a priori and a posteriori error bounds for the state,control and co-state variables.Numerical experiments are performed to validate the theoretical rates of convergence.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 spectral Galerkin approximation of optimal control problem governed by fractional elliptic equation is investigated.To deal with the nonlocality of fractional Laplacian operator the Caffarelli-Silvestre ...In this paper spectral Galerkin approximation of optimal control problem governed by fractional elliptic equation is investigated.To deal with the nonlocality of fractional Laplacian operator the Caffarelli-Silvestre extension is utilized.The first order optimality condition of the extended optimal control problem is derived.A spectral Galerkin discrete scheme for the extended problem based on weighted Laguerre polynomials is developed.A priori error estimates for the spectral Galerkin discrete scheme is proved.Numerical experiments are presented to show the effectiveness of our methods and to verify the theoretical findings.展开更多
In this paper,we investigate the Legendre Galerkin spectral approximation of quadratic optimal control problems governed by parabolic equations.A spectral approximation scheme for the parabolic optimal control problem...In this paper,we investigate the Legendre Galerkin spectral approximation of quadratic optimal control problems governed by parabolic equations.A spectral approximation scheme for the parabolic optimal control problem is presented.We obtain a posteriori error estimates of the approximated solutions for both the state and the control.展开更多
In this paper, we study numerical methods for an optimal control problem with pointwise state constraints. The traditional approaches often need to deal with the deltasingularity in the dual equation, which causes man...In this paper, we study numerical methods for an optimal control problem with pointwise state constraints. The traditional approaches often need to deal with the deltasingularity in the dual equation, which causes many difficulties in its theoretical analysis and numerical approximation. In our new approach we reformulate the state-constrained optimal control as a constrained minimization problems only involving the state, whose optimality condition is characterized by a fourth order elliptic variational inequality. Then direct numerical algorithms （nonconforming finite element approximation） are proposed for the inequality, and error estimates of the finite element approximation are derived. Numerical experiments illustrate the effectiveness of the new approach.展开更多
In this paper,optimize-then-discretize,variational discretization and the finite volume method are applied to solve the distributed optimal control problems governed by a second order hyperbolic equation.A semi-discre...In this paper,optimize-then-discretize,variational discretization and the finite volume method are applied to solve the distributed optimal control problems governed by a second order hyperbolic equation.A semi-discrete optimal system is obtained.We prove the existence and uniqueness of the solution to the semidiscrete optimal system and obtain the optimal order error estimates in L ∞(J;L 2)-and L ∞(J;H 1)-norm.Numerical experiments are presented to test these theoretical results.展开更多
In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Benard type under the pointwise control constraint. The states and the co-states are approximat...In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Benard type under the pointwise control constraint. The states and the co-states are approximated by polynomial functions of lowest-order mixed finite element space or piecewise linear functions and the control is approximated by piecewise constant functions. We give the superconvergence analysis for the control; it is proved that the approximation has a second-order rate of convergence. We further give the superconvergence analysis for the states and the co-states. Then we derive error estimates in L^∞-norm and optimal error estimates in L^2-norm.展开更多
In this paper,we investigate a priori error estimates for the quadratic optimal control problems governed by semilinear elliptic partial differential equations using higher order triangular mixed finite element method...In this paper,we investigate a priori error estimates for the quadratic optimal control problems governed by semilinear elliptic partial differential equations using higher order triangular mixed finite element methods.The state and the co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k(k≥0).A priori error estimates for the mixed finite element approximation of semilinear control problems are obtained.Finally,we present some numerical examples which confirm our theoretical results.展开更多
In this paper,we will investigate the error estimates and the superconvergence property of mixed finite element methods for a semilinear elliptic control problem with an integral constraint on control.The state and co...In this paper,we will investigate the error estimates and the superconvergence property of mixed finite element methods for a semilinear elliptic control problem with an integral constraint on control.The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element and the control variable is approximated by piecewise constant functions.We derive some superconvergence properties for the control variable and the state variables.Moreover,we derive L∞-and H−1-error estimates both for the control variable and the state variables.Finally,a numerical example is given to demonstrate the theoretical results.展开更多
We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization...We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.展开更多
Superconvergence and recovery a posteriori error estimates of the finite element ap- proximation for general convex optimal control problems are investigated in this paper. We obtain the superconvergence properties of...Superconvergence and recovery a posteriori error estimates of the finite element ap- proximation for general convex optimal control problems are investigated in this paper. We obtain the superconvergence properties of finite element solutions, and by using the superconvergence results we get recovery a posteriori error estimates which are asymptotically exact under some regularity conditions. Some numerical examples are provided to verify the theoretical results.展开更多
文摘In this paper, the optimal control problem of parabolic integro-differential equations is solved by gradient recovery based two-grid finite element method. Piecewise linear functions are used to approximate state and co-state variables, and piecewise constant function is used to approximate control variables. Generally, the optimal conditions for the problem are solved iteratively until the control variable reaches error tolerance. In order to calculate all the variables individually and parallelly, we introduce a gradient recovery based two-grid method. First, we solve the small scaled optimal control problem on coarse grids. Next, we use the gradient recovery technique to recover the gradients of state and co-state variables. Finally, using the recovered variables, we solve the large scaled optimal control problem for all variables independently. Moreover, we estimate priori error for the proposed scheme, and use an example to validate the theoretical results.
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In the optimal control problem of nonlinear dynamical system,the Hamiltonian formulation is useful and powerful to solve an optimal control force.However,the resulting Euler-Lagrange equations are not easy to solve,when the performance index is complicated,because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations.To be a numerical method,it is hard to exactly preserve all the specified conditions,which might deteriorate the accuracy of numerical solution.With this in mind,we develop a novel algorithm to find the solution of the optimal control problem of nonlinear Duffing oscillator,which can exactly satisfy all the required conditions for the minimality of the performance index.A new idea of shape functions method(SFM)is introduced,from which we can transform the optimal control problems to the initial value problems for the new variables,whose initial values are given arbitrarily,and meanwhile the terminal values are determined iteratively.Numerical examples confirm the high-performance of the iterative algorithms based on the SFM,which are convergence fast,and also provide very accurate solutions.The new algorithm is robust,even large noise is imposed on the input data.
文摘In this paper, we consider a fully discrete finite element approximation for time fractional optimal control problems. The state and adjoint state are approximated by triangular linear fi nite elements in space and <em>L</em>1 scheme in time. The control is obtained by the variational discretization technique. The main purpose of this work is to derive the convergence and superconvergence. A numerical example is presented to validate our theoretical results.
基金This work was partially supported by the NutionalNatural Science Foundation of China
文摘This paper deals with the optimal control problems of systems governed by a parabolic variational inequality coupled with a semilinear parabolic differential equations. The maximum principle and some kind of approximate controllability are studied.
文摘A kind of direct methods is presented for the solution of optimal control problems with state constraints. These methods are sequential quadratic programming methods. At every iteration a quadratic programming which is obtained by quadratic approximation to Lagrangian function and linear approximations to constraints is solved to get a search direction for a merit function. The merit function is formulated by augmenting the Lagrangian function with a penalty term. A line search is carried out along the search direction to determine a step length such that the merit function is decreased. The methods presented in this paper include continuous sequential quadratic programming methods and discreate sequential quadratic programming methods.
基金supported by the National Natural Science Foundation of China(Nos.11701253,11971259,11801216)Natural Science Foundation of Shandong Province(No.ZR2017BA010)。
文摘In this paper,we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random coefficients.There are two contributions of this paper.Firstly,we establish a scheme to approximate the optimality system by using the finite volume element method in the physical space and the meshfree method in the probability space,which is competitive for high-dimensional random inputs.Secondly,the a priori error estimates are derived for the state,the co-state and the control variables.Some numerical tests are carried out to confirm the theoretical results and demonstrate the efficiency of the proposed method.
文摘This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain.Such problems usually possess low regularity in the state variable due to the presence of measure data and the nonconvex nature of the domain.The low regularity of the solution allows the finite element approximations to converge at lower orders.We prove the existence,uniqueness and regularity results for the solution to the control problem satisfying the first order optimality condition.For our error analysis we have used piecewise linear elements for the approximation of the state and co-state variables,whereas piecewise constant functions are employed to approximate the control variable.The temporal discretization is based on the implicit Euler scheme.We derive both a priori and a posteriori error bounds for the state,control and co-state variables.Numerical experiments are performed to validate the theoretical rates of convergence.
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supported by the National Natural Science Foundation of China Project(Nos.12071402,11931003,12261131501,and 11971276)the Project of Scientific Research Fund of the Hunan Provincial Science and Technology Department(No.2022RC3022).
文摘In this paper spectral Galerkin approximation of optimal control problem governed by fractional elliptic equation is investigated.To deal with the nonlocality of fractional Laplacian operator the Caffarelli-Silvestre extension is utilized.The first order optimality condition of the extended optimal control problem is derived.A spectral Galerkin discrete scheme for the extended problem based on weighted Laguerre polynomials is developed.A priori error estimates for the spectral Galerkin discrete scheme is proved.Numerical experiments are presented to show the effectiveness of our methods and to verify the theoretical findings.
基金the National Basic Research Programthe National Natural Science Foundation of China(Grant No.2005CB321703)+2 种基金Scientific Research Fund of Hunan Provincial Education Departmentthe Outstanding Youth Scientist of the National Natural Science Foundation of China(Grant No.10625106)the National Basic Research Program of China(Grant No.2005CB321701)
文摘In this paper,we investigate the Legendre Galerkin spectral approximation of quadratic optimal control problems governed by parabolic equations.A spectral approximation scheme for the parabolic optimal control problem is presented.We obtain a posteriori error estimates of the approximated solutions for both the state and the control.
基金the National Natural Science Foundation of China (No.60474027 and 10771211)the National Basic Research Program under the Grant 2005CB321701
文摘In this paper, we study numerical methods for an optimal control problem with pointwise state constraints. The traditional approaches often need to deal with the deltasingularity in the dual equation, which causes many difficulties in its theoretical analysis and numerical approximation. In our new approach we reformulate the state-constrained optimal control as a constrained minimization problems only involving the state, whose optimality condition is characterized by a fourth order elliptic variational inequality. Then direct numerical algorithms （nonconforming finite element approximation） are proposed for the inequality, and error estimates of the finite element approximation are derived. Numerical experiments illustrate the effectiveness of the new approach.
基金supported by National Natural Science Foundation of China(Grant Nos.11261011,11271145 and 11031006)Foundation of Guizhou Science and Technology Department(Grant No.[2011]2098)+2 种基金Foundation for Talent Introduction of Guangdong Provincial UniversitySpecialized Research Fund for the Doctoral Program of Higher Education(Grant No. 20114407110009)the Project of Department of Education of Guangdong Province(Grant No. 2012KJCX0036)
文摘In this paper,optimize-then-discretize,variational discretization and the finite volume method are applied to solve the distributed optimal control problems governed by a second order hyperbolic equation.A semi-discrete optimal system is obtained.We prove the existence and uniqueness of the solution to the semidiscrete optimal system and obtain the optimal order error estimates in L ∞(J;L 2)-and L ∞(J;H 1)-norm.Numerical experiments are presented to test these theoretical results.
基金the Research Fund for Doctoral Program of High Education by China State Education Ministry under the Grant 2005042203
文摘In this paper, we consider the finite element approximation of the distributed optimal control problems of the stationary Benard type under the pointwise control constraint. The states and the co-states are approximated by polynomial functions of lowest-order mixed finite element space or piecewise linear functions and the control is approximated by piecewise constant functions. We give the superconvergence analysis for the control; it is proved that the approximation has a second-order rate of convergence. We further give the superconvergence analysis for the states and the co-states. Then we derive error estimates in L^∞-norm and optimal error estimates in L^2-norm.
基金supported by the Foundation for Talent Introduction of Guangdong Provincial Universities and CollegesPearl River Scholar Funded Scheme(2008)National Science Foundation of China(10971074).
文摘In this paper,we investigate a priori error estimates for the quadratic optimal control problems governed by semilinear elliptic partial differential equations using higher order triangular mixed finite element methods.The state and the co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k(k≥0).A priori error estimates for the mixed finite element approximation of semilinear control problems are obtained.Finally,we present some numerical examples which confirm our theoretical results.
基金supported by National Science Foundation of China(11271145)Foundation for Talent Introduction of Guangdong Provincial University,Specialized Research Fund for the Doctoral Program of Higher Education(20114407110009)+1 种基金the Project of Department of Education of Guangdong Province(2012KJCX0036)supported by China Postdoctoral Science Foundation funded project(2013M542188).
文摘In this paper,we will investigate the error estimates and the superconvergence property of mixed finite element methods for a semilinear elliptic control problem with an integral constraint on control.The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element and the control variable is approximated by piecewise constant functions.We derive some superconvergence properties for the control variable and the state variables.Moreover,we derive L∞-and H−1-error estimates both for the control variable and the state variables.Finally,a numerical example is given to demonstrate the theoretical results.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant No. 11271145）, the Foundation for Talent Introduction of Guangdong Provincial University, the Specialized Research Fund for the Doctoral Program of Higher Education （20114407110009）, and the Project of Department of Education of Guangdong Province （2012KJCX0036）.
文摘We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our 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 06A069Guangxi Natural Science Foundation 0575029
文摘Superconvergence and recovery a posteriori error estimates of the finite element ap- proximation for general convex optimal control problems are investigated in this paper. We obtain the superconvergence properties of finite element solutions, and by using the superconvergence results we get recovery a posteriori error estimates which are asymptotically exact under some regularity conditions. Some numerical examples are provided to verify the theoretical results.
