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
We compare 13 different a posteriori error estimators for the Poisson problem with lowest-order finite element discretization. Residual-based error estimators compete with a wide range of averaging estimators and esti...We compare 13 different a posteriori error estimators for the Poisson problem with lowest-order finite element discretization. Residual-based error estimators compete with a wide range of averaging estimators and estimators based on local problems. Among our five benchmark problems we also look on two examples with discontinuous isotropic diffusion and their impact on the performance of the estimators. （Supported by DFG Research Center MATHEON.）展开更多
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
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partly supported by the WCU program through KOSEF (R31-2008-000-10049-0)Supported by DFG Research Center MATHEON
文摘We compare 13 different a posteriori error estimators for the Poisson problem with lowest-order finite element discretization. Residual-based error estimators compete with a wide range of averaging estimators and estimators based on local problems. Among our five benchmark problems we also look on two examples with discontinuous isotropic diffusion and their impact on the performance of the estimators. （Supported by DFG Research Center MATHEON.）
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.
