In this paper,we consider the electromagnetic wave scattering problem from a periodic chiral structure.The scattering problem is simplified to a two-dimensional problem,and is discretized by a finite volume method com...In this paper,we consider the electromagnetic wave scattering problem from a periodic chiral structure.The scattering problem is simplified to a two-dimensional problem,and is discretized by a finite volume method combined with the perfectly matched layer(PML)technique.A residual-type a posteriori error estimate of the PML finite volume method is analyzed and the upper and lower bounds on the error are established in the H^(1)-norm.The crucial part of the a posteriori error analysis is to derive the error representation formula and use a L^(2)-orthogonality property of the residual which plays a similar role as the Galerkin orthogonality.An adaptive PML finite volume method is proposed to solve the scattering problem.The PML parameters such as the thickness of the layer and the medium property are determined through sharp a posteriori error estimate.Finally,numerical experiments are presented to illustrate the efficiency of the proposed method.展开更多
This paper extends the unifying theory for a posteriori error analysis of the nonconformingfinite element methods to the second order elliptic eigenvalue problem.In particular,the authorproposes the a posteriori error...This paper extends the unifying theory for a posteriori error analysis of the nonconformingfinite element methods to the second order elliptic eigenvalue problem.In particular,the authorproposes the a posteriori error estimator for nonconforming methods of the eigenvalue problems andprove its reliability and efficiency based on two assumptions concerning both the weak continuity andthe weak orthogonality of the nonconforming finite element spaces,respectively.In addition,the authorexamines these two assumptions for those nonconforming methods checked in literature for the Laplace,Stokes,and the linear elasticity problems.展开更多
The uniaxial perfectly matched layer (PML) method uses rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in deal- ing with problems involving anisotropic scatterers.In t...The uniaxial perfectly matched layer (PML) method uses rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in deal- ing with problems involving anisotropic scatterers.In this paper an adaptive uniaxial PML technique for solving the time harmonic Helmholtz scattering problem is devel- oped.The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates.The adaptive finite element method based on a posteriori error estimate is proposed to solve the PML equa- tion which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the PML absorb- ing layer.Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.In particular,it is demonstrated that the PML layer can be chosen as close to one wave-length from the scatterer and still yields good accuracy and efficiency in approximating the far fields.展开更多
An adaptive mixed least squares Galerkin/Petrov finite element method （FEM） is developed for stationary conduction convection problems. The mixed least squares Galerkin/Petrov FEM is consistent and stable for any co...An adaptive mixed least squares Galerkin/Petrov finite element method （FEM） is developed for stationary conduction convection problems. The mixed least squares Galerkin/Petrov FEM is consistent and stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. Using the general theory of Verfiirth, the posteriori error estimates of the residual type are derived. Finally, numerical tests are presented to illustrate the effectiveness of the method.展开更多
This work is concerned with time stepping finite element methods for abstract second order evolution problems. We derive optimal order a posteriori error estimates and a posteriori nodal superconvergence error estimat...This work is concerned with time stepping finite element methods for abstract second order evolution problems. We derive optimal order a posteriori error estimates and a posteriori nodal superconvergence error estimates using the energy approach and the duality argument. With the help of the a posteriori error estimator developed in this work, we will further propose an adaptive time stepping strategy. A number of numerical experiments are performed to illustrate the reliability and efficiency of the a posteriori error estimates and to assess the effectiveness of the proposed adaptive time stepping method.展开更多
In this paper, we investigate the a priori and a posteriori error estimates for the discontinuous Galerkin finite element approximation to a regularization version of the variational inequality of the second kind. We ...In this paper, we investigate the a priori and a posteriori error estimates for the discontinuous Galerkin finite element approximation to a regularization version of the variational inequality of the second kind. We show the optimal error estimates in the DG-norm （stronger than the H1 norm） and the L2 norm, respectively. Furthermore, some residual-based a posteriori error estimators are established which provide global upper bounds and local lower bounds on the discretization error. These a posteriori analysis results can be applied to develop the adaptive DG methods.展开更多
The electromagnetic wave propagation in the chiral medium is governed by Maxwell＇s equations together with the Drude-Born-Fedorov （constitutive） equations. The problem is simplified to a two-dimensional scattering ...The electromagnetic wave propagation in the chiral medium is governed by Maxwell＇s equations together with the Drude-Born-Fedorov （constitutive） equations. The problem is simplified to a two-dimensional scattering problem, and is formulated in a bounded domain by introducing two pairs of transparent boundary conditions. An a posteriori error estimate associated with the truncation of the nonlocal boundary operators is established. Based on the a posteriori error control, a finite element adaptive strategy is presented for computing the diffraction problem. The truncation parameter is determined through sharp a posteriori error estimate. Numerical experiments are included to illustrate the robustness and effectiveness of our error estimate and the proposed adaptive algorithm.展开更多
基金supported in part by the NSF of China(Grant No.12171141)by the Science and Technology Attack Plan Project of Henan province(Grant No.222102210049)supported in part by the NSF of China(Grant No.62102134).
文摘In this paper,we consider the electromagnetic wave scattering problem from a periodic chiral structure.The scattering problem is simplified to a two-dimensional problem,and is discretized by a finite volume method combined with the perfectly matched layer(PML)technique.A residual-type a posteriori error estimate of the PML finite volume method is analyzed and the upper and lower bounds on the error are established in the H^(1)-norm.The crucial part of the a posteriori error analysis is to derive the error representation formula and use a L^(2)-orthogonality property of the residual which plays a similar role as the Galerkin orthogonality.An adaptive PML finite volume method is proposed to solve the scattering problem.The PML parameters such as the thickness of the layer and the medium property are determined through sharp a posteriori error estimate.Finally,numerical experiments are presented to illustrate the efficiency of the proposed method.
文摘This paper extends the unifying theory for a posteriori error analysis of the nonconformingfinite element methods to the second order elliptic eigenvalue problem.In particular,the authorproposes the a posteriori error estimator for nonconforming methods of the eigenvalue problems andprove its reliability and efficiency based on two assumptions concerning both the weak continuity andthe weak orthogonality of the nonconforming finite element spaces,respectively.In addition,the authorexamines these two assumptions for those nonconforming methods checked in literature for the Laplace,Stokes,and the linear elasticity problems.
文摘The uniaxial perfectly matched layer (PML) method uses rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in deal- ing with problems involving anisotropic scatterers.In this paper an adaptive uniaxial PML technique for solving the time harmonic Helmholtz scattering problem is devel- oped.The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates.The adaptive finite element method based on a posteriori error estimate is proposed to solve the PML equa- tion which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the PML absorb- ing layer.Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.In particular,it is demonstrated that the PML layer can be chosen as close to one wave-length from the scatterer and still yields good accuracy and efficiency in approximating the far fields.
基金supported by the National Natural Science Foundation of China(Nos.10871156 and 11171269)the Fund of Xi'an Jiaotong University(No.2009xjtujc30)
文摘An adaptive mixed least squares Galerkin/Petrov finite element method （FEM） is developed for stationary conduction convection problems. The mixed least squares Galerkin/Petrov FEM is consistent and stable for any combination of discrete velocity and pressure spaces without requiring the Babuska-Brezzi stability condition. Using the general theory of Verfiirth, the posteriori error estimates of the residual type are derived. Finally, numerical tests are presented to illustrate the effectiveness of the method.
基金supported by National Natural Science Foundation of China(Grant Nos.1117121911161130004 and 11101199)+1 种基金E-Institutes of Shanghai Municipal Education Commission(Grant No.E03004)Program for New Century Excellent Talents in Fujian Province University(Grant No.JA12260)
文摘This work is concerned with time stepping finite element methods for abstract second order evolution problems. We derive optimal order a posteriori error estimates and a posteriori nodal superconvergence error estimates using the energy approach and the duality argument. With the help of the a posteriori error estimator developed in this work, we will further propose an adaptive time stepping strategy. A number of numerical experiments are performed to illustrate the reliability and efficiency of the a posteriori error estimates and to assess the effectiveness of the proposed adaptive time stepping method.
基金Supported by the National Natural Science Foundation of China(No.11071033)
文摘In this paper, we investigate the a priori and a posteriori error estimates for the discontinuous Galerkin finite element approximation to a regularization version of the variational inequality of the second kind. We show the optimal error estimates in the DG-norm （stronger than the H1 norm） and the L2 norm, respectively. Furthermore, some residual-based a posteriori error estimators are established which provide global upper bounds and local lower bounds on the discretization error. These a posteriori analysis results can be applied to develop the adaptive DG methods.
基金The work of the second author was supported by the NSFC （No. 11301267） and by the Natural Science Foundation for Colleges and Universities in Jiangsu Province （No.12KJB110007）.
文摘The electromagnetic wave propagation in the chiral medium is governed by Maxwell＇s equations together with the Drude-Born-Fedorov （constitutive） equations. The problem is simplified to a two-dimensional scattering problem, and is formulated in a bounded domain by introducing two pairs of transparent boundary conditions. An a posteriori error estimate associated with the truncation of the nonlocal boundary operators is established. Based on the a posteriori error control, a finite element adaptive strategy is presented for computing the diffraction problem. The truncation parameter is determined through sharp a posteriori error estimate. Numerical experiments are included to illustrate the robustness and effectiveness of our error estimate and the proposed adaptive algorithm.
