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
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.
