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
An adaptive multi-penalty discontinuous Galerkin method(AMPDG)for the diffusion problem is considered.Convergence and quasi-optimality of the AM-PDG are proved.Compared with the analyses for the adaptive finite elemen...An adaptive multi-penalty discontinuous Galerkin method(AMPDG)for the diffusion problem is considered.Convergence and quasi-optimality of the AM-PDG are proved.Compared with the analyses for the adaptive finite element method or the adaptive interior penalty discontinuous Galerkin method,extra works are done to overcome the difficulties caused by the additional penalty terms.展开更多
In this paper, we study some new systems of generalized quasi-variational inclusion problems in FC-spaces without convexity structure.By applying an existence theorem of maximal elements of set-valued mappings due to ...In this paper, we study some new systems of generalized quasi-variational inclusion problems in FC-spaces without convexity structure.By applying an existence theorem of maximal elements of set-valued mappings due to the author, some new existence theorems of solutions for the systems of generalized quasi-variational inclusion problems are proved in noncompact FC-spaces. As applications, some existence results of solutions for the system of quasi-optimization problems and mathematical programs with the systems of generalized quasi-variational inclusion constraints are obtained in FC-spaces.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.
基金This research was partially the National Natural Science Foundation of China under grants 11525103 and 91130004.
文摘An adaptive multi-penalty discontinuous Galerkin method(AMPDG)for the diffusion problem is considered.Convergence and quasi-optimality of the AM-PDG are proved.Compared with the analyses for the adaptive finite element method or the adaptive interior penalty discontinuous Galerkin method,extra works are done to overcome the difficulties caused by the additional penalty terms.
基金supported by the Scientific Research Fun of Sichuan Normal University(09ZDL04)the Sichuan Province Leading Academic Discipline Project(SZD0406)
文摘In this paper, we study some new systems of generalized quasi-variational inclusion problems in FC-spaces without convexity structure.By applying an existence theorem of maximal elements of set-valued mappings due to the author, some new existence theorems of solutions for the systems of generalized quasi-variational inclusion problems are proved in noncompact FC-spaces. As applications, some existence results of solutions for the system of quasi-optimization problems and mathematical programs with the systems of generalized quasi-variational inclusion constraints are obtained in FC-spaces.
