Convergence and Optimality of Adaptive Methods for Poisson's Equation in the FEEC Framework
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 two dimensions, 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.
Cite
@article{arxiv.1306.1886,
title = {Convergence and Optimality of Adaptive Methods for Poisson's Equation in the FEEC Framework},
author = {Michael Holst and Yuwen Li and Adam Mihalik and Ryan Szypowski},
journal= {arXiv preprint arXiv:1306.1886},
year = {2018}
}
Comments
20 pages, no figures. Version 3 contains improved proofs of the main results, some error corrections, and now also the optimality result