Volume 38, Issue 5
Convergence and Optimality of Adaptive Mixed Methods for Poisson's Equation in the FEEC Framework

J. Comp. Math., 38 (2020), pp. 748-767.

Published online: 2020-04

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• Abstract

Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther and others over the last decade to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can then be obtained by solving finite-dimensional subcomplex problems. Chen, Holst, and Xu (Math. Comp. 78 (2009) 35-53) established convergence and optimality of an adaptive mixed finite element method using Raviart-Thomas or Brezzi-Douglas-Marini elements for Poisson's equation on contractible domains in $\mathbb{R}^2$, which can be viewed as a boundary problem on the de Rham 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 results on domains with general topology and spatial dimension. In particular, we construct a reliable and efficient error estimator and a sharper quasi-orthogonality result using a novel technique. Without marking for data oscillation, our adaptive method is a contraction with respect to a total error incorporating the error estimator and data oscillation.

• Keywords

Finite element exterior calculus, Adaptive finite element methods, A posteriori error estimates, Convergence, Quasi-optimality.

65N12, 65N15, 65N30, 65N50

mholst@math.ucsd.edu (Michael Holst)

yul739@ucsd.edu (Yuwen Li)

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@Article{JCM-38-748, author = {Holst , Michael and Li , Yuwen and Mihalik , Adam and Szypowski , Ryan }, title = {Convergence and Optimality of Adaptive Mixed Methods for Poisson's Equation in the FEEC Framework}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {38}, number = {5}, pages = {748--767}, abstract = {

Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther and others over the last decade to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can then be obtained by solving finite-dimensional subcomplex problems. Chen, Holst, and Xu (Math. Comp. 78 (2009) 35-53) established convergence and optimality of an adaptive mixed finite element method using Raviart-Thomas or Brezzi-Douglas-Marini elements for Poisson's equation on contractible domains in $\mathbb{R}^2$, which can be viewed as a boundary problem on the de Rham 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 results on domains with general topology and spatial dimension. In particular, we construct a reliable and efficient error estimator and a sharper quasi-orthogonality result using a novel technique. Without marking for data oscillation, our adaptive method is a contraction with respect to a total error incorporating the error estimator and data oscillation.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1905-m2018-0265}, url = {http://global-sci.org/intro/article_detail/jcm/16668.html} }
TY - JOUR T1 - Convergence and Optimality of Adaptive Mixed Methods for Poisson's Equation in the FEEC Framework AU - Holst , Michael AU - Li , Yuwen AU - Mihalik , Adam AU - Szypowski , Ryan JO - Journal of Computational Mathematics VL - 5 SP - 748 EP - 767 PY - 2020 DA - 2020/04 SN - 38 DO - http://doi.org/10.4208/jcm.1905-m2018-0265 UR - https://global-sci.org/intro/article_detail/jcm/16668.html KW - Finite element exterior calculus, Adaptive finite element methods, A posteriori error estimates, Convergence, Quasi-optimality. AB -

Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther and others over the last decade to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can then be obtained by solving finite-dimensional subcomplex problems. Chen, Holst, and Xu (Math. Comp. 78 (2009) 35-53) established convergence and optimality of an adaptive mixed finite element method using Raviart-Thomas or Brezzi-Douglas-Marini elements for Poisson's equation on contractible domains in $\mathbb{R}^2$, which can be viewed as a boundary problem on the de Rham 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 results on domains with general topology and spatial dimension. In particular, we construct a reliable and efficient error estimator and a sharper quasi-orthogonality result using a novel technique. Without marking for data oscillation, our adaptive method is a contraction with respect to a total error incorporating the error estimator and data oscillation.

Michael Holst, Yuwen Li, Adam Mihalik & Ryan Szypowski. (2020). Convergence and Optimality of Adaptive Mixed Methods for Poisson's Equation in the FEEC Framework. Journal of Computational Mathematics. 38 (5). 748-767. doi:10.4208/jcm.1905-m2018-0265
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