Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther andothers over the last decade to exploit the observation that mixed variational problems canbe posed on a Hilbert complex, and Galerkin-t...Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther andothers over the last decade to exploit the observation that mixed variational problems canbe posed on a Hilbert complex, and Galerkin-type mixed methods can then be obtained bysolving finite-dimensional subcomplex problems. Chen, Holst, and Xu (Math. Comp. 78(2009) 35–53) established convergence and optimality of an adaptive mixed finite elementmethod using Raviart–Thomas or Brezzi–Douglas–Marini elements for Poisson’s equationon contractible domains in R^2, which can be viewed as a boundary problem on the deRham complex. Recently Demlow and Hirani (Found. Math. Comput. 14 (2014) 1337–1371) developed fundamental tools for a posteriori analysis on the de Rham complex.In this paper, we use tools in FEEC to construct convergence and complexity resultson domains with general topology and spatial dimension. In particular, we construct areliable and efficient error estimator and a sharper quasi-orthogonality result using a noveltechnique. Without marking for data oscillation, our adaptive method is a contractionwith respect to a total error incorporating the error estimator and data oscillation.展开更多
In this paper,ETD3-Padéand ETD4-PadéGalerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions.An ETD-based RK ...In this paper,ETD3-Padéand ETD4-PadéGalerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions.An ETD-based RK is used for time integration of the corresponding equation.To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator,the Padéapproach is used for such an exponential operator approximation,which in turn leads to the corresponding ETD-Padéschemes.An unconditional L^(2) numerical stability is proved for the proposed numerical schemes,under a global Lipshitz continuity assumption.In addition,optimal rate error estimates are provided,which gives the convergence order of O(k^(3)+h^(r))(ETD3-Padé)or O(k^(4)+h^(r))(ETD4-Padé)in the L^(2)norm,respectively.Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.展开更多
Abstract The main objective of this paper is to present a new rectangular nonconforming finite element scheme with the second order convergence behavior for approximation of Maxwell＇s equations. Then the correspondin...Abstract The main objective of this paper is to present a new rectangular nonconforming finite element scheme with the second order convergence behavior for approximation of Maxwell＇s equations. Then the corresponding optimal error estimates are derived. The difficulty in construction of this finite element scheme is how to choose a compatible pair of degrees of freedom and shape function space so as to make the consistency error due to the nonconformity of the element being of order O（h^3）, properly one order higher than that of its interpolation error O（h^2） in the broken energy norm, where h is the subdivision parameter tending to zero.展开更多
基金MS was partially supported by NSF Awards 1620366,1262982,and 1217175.YL was partially supported by NSF Award 1620366.AM adn RS were partially supported by NSF Award 1217175.
文摘Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther andothers over the last decade to exploit the observation that mixed variational problems canbe posed on a Hilbert complex, and Galerkin-type mixed methods can then be obtained bysolving finite-dimensional subcomplex problems. Chen, Holst, and Xu (Math. Comp. 78(2009) 35–53) established convergence and optimality of an adaptive mixed finite elementmethod using Raviart–Thomas or Brezzi–Douglas–Marini elements for Poisson’s equationon contractible domains in R^2, which can be viewed as a boundary problem on the deRham complex. Recently Demlow and Hirani (Found. Math. Comput. 14 (2014) 1337–1371) developed fundamental tools for a posteriori analysis on the de Rham complex.In this paper, we use tools in FEEC to construct convergence and complexity resultson domains with general topology and spatial dimension. In particular, we construct areliable and efficient error estimator and a sharper quasi-orthogonality result using a noveltechnique. Without marking for data oscillation, our adaptive method is a contractionwith respect to a total error incorporating the error estimator and data oscillation.
文摘In this paper,ETD3-Padéand ETD4-PadéGalerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions.An ETD-based RK is used for time integration of the corresponding equation.To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator,the Padéapproach is used for such an exponential operator approximation,which in turn leads to the corresponding ETD-Padéschemes.An unconditional L^(2) numerical stability is proved for the proposed numerical schemes,under a global Lipshitz continuity assumption.In addition,optimal rate error estimates are provided,which gives the convergence order of O(k^(3)+h^(r))(ETD3-Padé)or O(k^(4)+h^(r))(ETD4-Padé)in the L^(2)norm,respectively.Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.
基金Supported by the National Natural Science Foundation of China (No. 10971203)the Doctor Foundationof Henan Institute of Engineering (No. D09008)
文摘Abstract The main objective of this paper is to present a new rectangular nonconforming finite element scheme with the second order convergence behavior for approximation of Maxwell＇s equations. Then the corresponding optimal error estimates are derived. The difficulty in construction of this finite element scheme is how to choose a compatible pair of degrees of freedom and shape function space so as to make the consistency error due to the nonconformity of the element being of order O（h^3）, properly one order higher than that of its interpolation error O（h^2） in the broken energy norm, where h is the subdivision parameter tending to zero.
