Superconvergence and recovery type a posteriori error estimators are analyzed for Pian and Sumihara's 4-node hybrid stress quadrilateral finite element method for linear elasticity problems. Superconvergence of or...Superconvergence and recovery type a posteriori error estimators are analyzed for Pian and Sumihara's 4-node hybrid stress quadrilateral finite element method for linear elasticity problems. Superconvergence of order O(h^(1+min){α,1}) is established for both the displacement approximation in H^1-norm and the stress approximation in L^2-norm under a mesh assumption, where α > 0 is a parameter characterizing the distortion of meshes from parallelograms to quadrilaterals. Recovery type approximations for the displacement gradients and the stress tensor are constructed, and a posteriori error estimators based on the recovered quantities are shown to be asymptotically exact. Numerical experiments confirm the theoretical results.展开更多
This paper is devoted to the Polynomial Preserving Recovery (PPR) based a posteriori error analysis for the second-order elliptic non-symmetric eigenvalue problem. An asymptotically exact a posteriori error estimator ...This paper is devoted to the Polynomial Preserving Recovery (PPR) based a posteriori error analysis for the second-order elliptic non-symmetric eigenvalue problem. An asymptotically exact a posteriori error estimator is proposed for solving the convection-dominated non-symmetric eigenvalue problem with non-smooth eigenfunctions or multiple eigenvalues. Numerical examples confirm our theoretical analysis.展开更多
In this paper,we develop a residual-based a posteriori error estimator for the time-dependent Maxwell’s equations in the cold plasma.Here we consider a semi-discrete interior penalty discontinuous Galerkin(DG)method ...In this paper,we develop a residual-based a posteriori error estimator for the time-dependent Maxwell’s equations in the cold plasma.Here we consider a semi-discrete interior penalty discontinuous Galerkin(DG)method for solving the governing equations.We provide both the upper bound and lower bound analysis for the error estimator.This is the first posteriori error analysis carried out for the Maxwell’s equations in dispersive media.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 this paper,a posteriori error estimates for the generalized Schwartz method with Dirichlet boundary conditions on the interfaces for advection-diffusion equation with second order boundary value problems are proved...In this paper,a posteriori error estimates for the generalized Schwartz method with Dirichlet boundary conditions on the interfaces for advection-diffusion equation with second order boundary value problems are proved by using the Euler time scheme combined with Galerkin spatial method.Furthermore,an asymptotic behavior in Sobolev norm is de-duced using Benssoussau-Lions'algorithm.Finally,the results of some numerical experiments are presented to support the theory.展开更多
Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level met...Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.展开更多
A nonlinear Galerkin mixed element (NGME) method and a posteriori error exstimator based on the method are established for the stationary Navier-Stokes equations. The existence and error estimates of the NGME solution...A nonlinear Galerkin mixed element (NGME) method and a posteriori error exstimator based on the method are established for the stationary Navier-Stokes equations. The existence and error estimates of the NGME solution are first discussed, and then a posteriori error estimator based on the NGME method is derived.展开更多
The main aim of this paper is to give an anisotropic posteriori error estimator. We firstly study the convergence of bilineax finite element for the second order problem under anisotropic meshes. By using some novel a...The main aim of this paper is to give an anisotropic posteriori error estimator. We firstly study the convergence of bilineax finite element for the second order problem under anisotropic meshes. By using some novel approaches and techniques, the optimal error estimates and some superconvergence results axe obtained without the regulaxity assumption and quasi-uniform assumption requirements on the meshes. Then, based on these results, we give an anisotropic posteriori error estimate for the second problem.展开更多
In this paper, a posteriori error estimates were derived for piecewise linear finite element approximations to parabolic obstacle problems. The instrumental ingredient was introduced as a new interpolation operator wh...In this paper, a posteriori error estimates were derived for piecewise linear finite element approximations to parabolic obstacle problems. The instrumental ingredient was introduced as a new interpolation operator which has optimal approximation properties and preserves positivity. With the help of the interpolation operator the upper and lower bounds were obtained.展开更多
A posteriori error estimate of the discontinuous-streamline diffusion method for first-order hyperbolic equations was presented, which can be used to adjust space mesh reasonably. A numerical example is given to illus...A posteriori error estimate of the discontinuous-streamline diffusion method for first-order hyperbolic equations was presented, which can be used to adjust space mesh reasonably. A numerical example is given to illustrate the accuracy and feasibility of this method.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 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
In this paper,we present a posteriori error estimates of the weak Galerkin finite element method for the steady-state Poisson-Nernst-Planck equations.The a posteriori error estimators for the electrostatic potential a...In this paper,we present a posteriori error estimates of the weak Galerkin finite element method for the steady-state Poisson-Nernst-Planck equations.The a posteriori error estimators for the electrostatic potential and ion concentrations are constructed.The reliability and efficiency of the estimators are verified by the upper and lower bounds of the energy norm of the error.The a posteriori error estimators are applied to the adaptive weak Galerkin algorithm for triangle,quadrilateral and polygonal meshes with hanging nodes.Finally,numerical results demonstrate the effectiveness of the adaptive algorithm guided by our constructed estimators.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
In this paper,we study a posteriori error estimates of the L1 scheme for time discretizations of time fractional parabolic differential equations,whose solutions have generally the initial singularity.To derive optima...In this paper,we study a posteriori error estimates of the L1 scheme for time discretizations of time fractional parabolic differential equations,whose solutions have generally the initial singularity.To derive optimal order a posteriori error estimates,the quadratic reconstruction for the L1 method and the necessary fractional integral reconstruction for the first-step integration are introduced.By using these continuous,piecewise time reconstructions,the upper and lower error bounds depending only on the discretization parameters and the data of the problems are derived.Various numerical experiments for the one-dimensional linear fractional parabolic equations with smooth or nonsmooth exact solution are used to verify and complement our theoretical results,with the convergence ofαorder for the nonsmooth case on a uniform mesh.To recover the optimal convergence order 2-αon a nonuniform mesh,we further develop a time adaptive algorithm by means of barrier function recently introduced.The numerical implementations are performed on nonsmooth case again and verify that the true error and a posteriori error can achieve the optimal convergence order in adaptive mesh.展开更多
In this paper, we derive optimal order a posteriori error estimates for the local dis- continuous Galerkin （LDC） method for linear convection-diffusion problems in one space dimension. One of the key ingredients in ...In this paper, we derive optimal order a posteriori error estimates for the local dis- continuous Galerkin （LDC） method for linear convection-diffusion problems in one space dimension. One of the key ingredients in our analysis is the recent optimal superconver- gence result in [Y. Yang and C.-W. Shu, J. Comp. Math., 33 （2015）, pp. 323-340]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the L2-norm to （p ＋ 1）-degree right and left Radau interpolating polynomials under mesh re- finement. The order of convergence is proved to be p ＋ 2, when piecewise polynomials of degree at most p are used. These results are used to show that the leading error terms on each element for the solution and its derivative are proportional to （p ＋ 1）-degree right and left Radau polynomials. We further prove that, for smooth solutions, the a posteriori LDG error estimates, which were constructed by the author in an earlier paper, converge, at a fixed time, to the true spatial errors in the L2-norm at （.9（hp＋2） rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at （9（h） rate. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be p＋3/2 and 1/2, respectively. Our proofs are valid for arbitrary regular meshes using PP polynomials with p ≥ 1. Several numerical experiments are performed to validate the theoretical results.展开更多
This article is concerned with the numerical detection of bifurcation points of nonlinear partial differential equations as some parameter of interest is varied.In particular,we study in detail the numerical approxima...This article is concerned with the numerical detection of bifurcation points of nonlinear partial differential equations as some parameter of interest is varied.In particular,we study in detail the numerical approximation of the Bratu problem,based on exploiting the symmetric version of the interior penalty discontinuous Galerkin finite element method.A framework for a posteriori control of the discretization error in the computed critical parameter value is developed based upon the application of the dual weighted residual(DWR)approach.Numerical experiments are presented to highlight the practical performance of the proposed a posteriori error estimator.展开更多
In this paper,we study the a posteriori error estimator of SDG method for variable coefficients time-harmonic Maxwell's equations.We propose two a posteriori error estimators,one is the recovery-type estimator,and...In this paper,we study the a posteriori error estimator of SDG method for variable coefficients time-harmonic Maxwell's equations.We propose two a posteriori error estimators,one is the recovery-type estimator,and the other is the residual-type estimator.We first propose the curl-recovery method for the staggered discontinuous Galerkin method(SDGM),and based on the super-convergence result of the postprocessed solution,an asymptotically exact error estimator is constructed.The residual-type a posteriori error estimator is also proposed,and it's reliability and effectiveness are proved for variable coefficients time-harmonic Maxwell's equations.The efficiency and robustness of the proposed estimators is demonstrated by the numerical experiments.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
Four primal discontinuous Galerkin methods are applied to solve reactive transport problems, namely, Oden-BabuSka-Baumann DG （OBB-DG）, non-symmetric interior penalty Galerkin （NIPG）, symmetric interior penalty Gal...Four primal discontinuous Galerkin methods are applied to solve reactive transport problems, namely, Oden-BabuSka-Baumann DG （OBB-DG）, non-symmetric interior penalty Galerkin （NIPG）, symmetric interior penalty Galerkin （SIPG）, and incomplete interior penalty Galerkin （IIPG）. A unified a posteriori residual-type error estimation is derived explicitly for these methods. From the computed solution and given data, explicit estimators can be computed efficiently and directly, which can be used as error indicators for adaptation. Unlike in the reference [10], we obtain the error estimators in L^2 （L^2） norm by using duality techniques instead of in L^2（H^1） norm.展开更多
基金supported by National Natural Science Foundation of China (Grant No. 11171239)Major Research Plan of National Natural Science Foundation of China (Grant No. 91430105)
文摘Superconvergence and recovery type a posteriori error estimators are analyzed for Pian and Sumihara's 4-node hybrid stress quadrilateral finite element method for linear elasticity problems. Superconvergence of order O(h^(1+min){α,1}) is established for both the displacement approximation in H^1-norm and the stress approximation in L^2-norm under a mesh assumption, where α > 0 is a parameter characterizing the distortion of meshes from parallelograms to quadrilaterals. Recovery type approximations for the displacement gradients and the stress tensor are constructed, and a posteriori error estimators based on the recovered quantities are shown to be asymptotically exact. Numerical experiments confirm the theoretical results.
基金Supported by the National Natural Science Foundation of China (Grant Nos.1236108412001130)。
文摘This paper is devoted to the Polynomial Preserving Recovery (PPR) based a posteriori error analysis for the second-order elliptic non-symmetric eigenvalue problem. An asymptotically exact a posteriori error estimator is proposed for solving the convection-dominated non-symmetric eigenvalue problem with non-smooth eigenfunctions or multiple eigenvalues. Numerical examples confirm our theoretical analysis.
基金supported by National Science Foundation grant DMS-0810896。
文摘In this paper,we develop a residual-based a posteriori error estimator for the time-dependent Maxwell’s equations in the cold plasma.Here we consider a semi-discrete interior penalty discontinuous Galerkin(DG)method for solving the governing equations.We provide both the upper bound and lower bound analysis for the error estimator.This is the first posteriori error analysis carried out for the Maxwell’s equations in dispersive media.
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 this paper,a posteriori error estimates for the generalized Schwartz method with Dirichlet boundary conditions on the interfaces for advection-diffusion equation with second order boundary value problems are proved by using the Euler time scheme combined with Galerkin spatial method.Furthermore,an asymptotic behavior in Sobolev norm is de-duced using Benssoussau-Lions'algorithm.Finally,the results of some numerical experiments are presented to support the theory.
文摘Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.
文摘A nonlinear Galerkin mixed element (NGME) method and a posteriori error exstimator based on the method are established for the stationary Navier-Stokes equations. The existence and error estimates of the NGME solution are first discussed, and then a posteriori error estimator based on the NGME method is derived.
文摘The main aim of this paper is to give an anisotropic posteriori error estimator. We firstly study the convergence of bilineax finite element for the second order problem under anisotropic meshes. By using some novel approaches and techniques, the optimal error estimates and some superconvergence results axe obtained without the regulaxity assumption and quasi-uniform assumption requirements on the meshes. Then, based on these results, we give an anisotropic posteriori error estimate for the second problem.
基金Project supported by National Natural Science Foundation ofChina (Grant No .10471089)
文摘In this paper, a posteriori error estimates were derived for piecewise linear finite element approximations to parabolic obstacle problems. The instrumental ingredient was introduced as a new interpolation operator which has optimal approximation properties and preserves positivity. With the help of the interpolation operator the upper and lower bounds were obtained.
文摘A posteriori error estimate of the discontinuous-streamline diffusion method for first-order hyperbolic equations was presented, which can be used to adjust space mesh reasonably. A numerical example is given to illustrate the accuracy and feasibility of this method.
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 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supported by the National Natural Science Foundation of China(Grant No.12471363).
文摘In this paper,we present a posteriori error estimates of the weak Galerkin finite element method for the steady-state Poisson-Nernst-Planck equations.The a posteriori error estimators for the electrostatic potential and ion concentrations are constructed.The reliability and efficiency of the estimators are verified by the upper and lower bounds of the energy norm of the error.The a posteriori error estimators are applied to the adaptive weak Galerkin algorithm for triangle,quadrilateral and polygonal meshes with hanging nodes.Finally,numerical results demonstrate the effectiveness of the adaptive algorithm guided by our constructed estimators.
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supported by the Natural Science Foundation of China(Grants 12271367,12071403)by the Shanghai Science and Technology Planning Projects(Grant 20JC1414200).
文摘In this paper,we study a posteriori error estimates of the L1 scheme for time discretizations of time fractional parabolic differential equations,whose solutions have generally the initial singularity.To derive optimal order a posteriori error estimates,the quadratic reconstruction for the L1 method and the necessary fractional integral reconstruction for the first-step integration are introduced.By using these continuous,piecewise time reconstructions,the upper and lower error bounds depending only on the discretization parameters and the data of the problems are derived.Various numerical experiments for the one-dimensional linear fractional parabolic equations with smooth or nonsmooth exact solution are used to verify and complement our theoretical results,with the convergence ofαorder for the nonsmooth case on a uniform mesh.To recover the optimal convergence order 2-αon a nonuniform mesh,we further develop a time adaptive algorithm by means of barrier function recently introduced.The numerical implementations are performed on nonsmooth case again and verify that the true error and a posteriori error can achieve the optimal convergence order in adaptive mesh.
文摘In this paper, we derive optimal order a posteriori error estimates for the local dis- continuous Galerkin （LDC） method for linear convection-diffusion problems in one space dimension. One of the key ingredients in our analysis is the recent optimal superconver- gence result in [Y. Yang and C.-W. Shu, J. Comp. Math., 33 （2015）, pp. 323-340]. We first prove that the LDG solution and its spatial derivative, respectively, converge in the L2-norm to （p ＋ 1）-degree right and left Radau interpolating polynomials under mesh re- finement. The order of convergence is proved to be p ＋ 2, when piecewise polynomials of degree at most p are used. These results are used to show that the leading error terms on each element for the solution and its derivative are proportional to （p ＋ 1）-degree right and left Radau polynomials. We further prove that, for smooth solutions, the a posteriori LDG error estimates, which were constructed by the author in an earlier paper, converge, at a fixed time, to the true spatial errors in the L2-norm at （.9（hp＋2） rate. Finally, we prove that the global effectivity indices in the L2-norm converge to unity at （9（h） rate. These results improve upon our previously published work in which the order of convergence for the a posteriori error estimates and the global effectivity index are proved to be p＋3/2 and 1/2, respectively. Our proofs are valid for arbitrary regular meshes using PP polynomials with p ≥ 1. Several numerical experiments are performed to validate the theoretical results.
基金the financial support of the EPSRC under the grant EP/E013724the support of the EPSRC under the grant EP/F01340X.
文摘This article is concerned with the numerical detection of bifurcation points of nonlinear partial differential equations as some parameter of interest is varied.In particular,we study in detail the numerical approximation of the Bratu problem,based on exploiting the symmetric version of the interior penalty discontinuous Galerkin finite element method.A framework for a posteriori control of the discretization error in the computed critical parameter value is developed based upon the application of the dual weighted residual(DWR)approach.Numerical experiments are presented to highlight the practical performance of the proposed a posteriori error estimator.
基金supported by NSFC Projects(Nos.11771371,12171411,11971410)Project of Scientific Research Fund of Hunan Provincial Science and Technology Department(No.2018WK4006)+1 种基金Project of Scientific Research Fund of Hunan Provincial Science and Technology Department,China(No.2020ZYT003)National defense basic scientific research program JCKY2019403D001.
文摘In this paper,we study the a posteriori error estimator of SDG method for variable coefficients time-harmonic Maxwell's equations.We propose two a posteriori error estimators,one is the recovery-type estimator,and the other is the residual-type estimator.We first propose the curl-recovery method for the staggered discontinuous Galerkin method(SDGM),and based on the super-convergence result of the postprocessed solution,an asymptotically exact error estimator is constructed.The residual-type a posteriori error estimator is also proposed,and it's reliability and effectiveness are proved for variable coefficients time-harmonic Maxwell's equations.The efficiency and robustness of the proposed estimators is demonstrated by the numerical experiments.
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This work is supported by Program for New Century Excellent Talents in University of China State Education Ministry NCET-04-0776, National Science Foundation of China, the National Basic Research Program under the Grant 2005CB321703, and the key project of China State Education Ministry and Hunan Education Commission.
文摘Four primal discontinuous Galerkin methods are applied to solve reactive transport problems, namely, Oden-BabuSka-Baumann DG （OBB-DG）, non-symmetric interior penalty Galerkin （NIPG）, symmetric interior penalty Galerkin （SIPG）, and incomplete interior penalty Galerkin （IIPG）. A unified a posteriori residual-type error estimation is derived explicitly for these methods. From the computed solution and given data, explicit estimators can be computed efficiently and directly, which can be used as error indicators for adaptation. Unlike in the reference [10], we obtain the error estimators in L^2 （L^2） norm by using duality techniques instead of in L^2（H^1） norm.
