This paper aims at a general guideline to obtain a posteriori error estimates for the finite element error control in computational partial differential equations.In the abstract setting of mixed formulations,a genera...This paper aims at a general guideline to obtain a posteriori error estimates for the finite element error control in computational partial differential equations.In the abstract setting of mixed formulations,a generalised formulation of the corresponding residuals is proposed which then allows for the unified estimation of the respective dual norms.Notably,this can be done with an approach which is applicable in the same way to conforming,nonconforming and mixed discretisations.Subsequently,the unified approach is applied to various model problems.In particular,we consider the Laplace,Stokes,Navier-Lamé,and the semi-discrete eddy current equations.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
We develop and analyze an adaptive hybridized Interior Penalty Discontinuous Galerkin(IPDG-H)method for H(curl)-elliptic boundary value problems in 2D or 3D arising from a semi-discretization of the eddy currents equ...We develop and analyze an adaptive hybridized Interior Penalty Discontinuous Galerkin(IPDG-H)method for H(curl)-elliptic boundary value problems in 2D or 3D arising from a semi-discretization of the eddy currents equations.The method can be derived from a mixed formulation of the given boundary value problem and involves a Lagrange multiplier that is an approximation of the tangential traces of the primal variable on the interfaces of the underlying triangulation of the computational domain.It is shown that the IPDG-H technique can be equivalently formulated and thus implemented as a mortar method.The mesh adaptation is based on a residual-type a posteriori error estimator consisting of element and face residuals.Within a unified framework for adaptive finite element methods,we prove the reliability of the estimator up to a consistency error.The performance of the adaptive symmetric IPDG-H method is documented by numerical results for representative test examples in 2D.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 by the German National Science Foundation DFG within the Research Center MATHEON and,with the first author,by the WCU program through KOSEF(R31-2008-000-10049-0)support by the NSF grants DMS-0707602,DMS-0810176,DMS-0811153,DMS-0914788+2 种基金the German National Science Foundation within the Priority Programs SPP 1253,SPP 1506the German Federal Ministry for Education and Research(BMBF)within the projects BMBF-FROPT and BMBF-MeFreSimthe European Science Foundation(ESF)within the ESF Program OPTPDE.
文摘This paper aims at a general guideline to obtain a posteriori error estimates for the finite element error control in computational partial differential equations.In the abstract setting of mixed formulations,a generalised formulation of the corresponding residuals is proposed which then allows for the unified estimation of the respective dual norms.Notably,this can be done with an approach which is applicable in the same way to conforming,nonconforming and mixed discretisations.Subsequently,the unified approach is applied to various model problems.In particular,we consider the Laplace,Stokes,Navier-Lamé,and the semi-discrete eddy current equations.
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The work of the first author has been supported by the German Na-tional Science Foundation DFG within the Research Center MATHEON and by the WCU program through KOSEF(R31-2008-000-10049-0).The other authors acknowledge sup-port by the NSF grant DMS-0810176.1
文摘We develop and analyze an adaptive hybridized Interior Penalty Discontinuous Galerkin(IPDG-H)method for H(curl)-elliptic boundary value problems in 2D or 3D arising from a semi-discretization of the eddy currents equations.The method can be derived from a mixed formulation of the given boundary value problem and involves a Lagrange multiplier that is an approximation of the tangential traces of the primal variable on the interfaces of the underlying triangulation of the computational domain.It is shown that the IPDG-H technique can be equivalently formulated and thus implemented as a mortar method.The mesh adaptation is based on a residual-type a posteriori error estimator consisting of element and face residuals.Within a unified framework for adaptive finite element methods,we prove the reliability of the estimator up to a consistency error.The performance of the adaptive symmetric IPDG-H method is documented by numerical results for representative test examples in 2D.
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.