Convergence and Optimality of Adaptive Methods for Poisson's Equation in the FEEC Framework

TL;DR

This paper advances the theoretical understanding of adaptive finite element methods for Poisson's equation within the FEEC framework, establishing convergence and optimality on complex domains using new error estimation techniques.

Contribution

It introduces a reliable and efficient a posteriori error estimator and proves convergence and optimality of adaptive methods on general topological domains with novel quasi-orthogonality results.

Findings

Convergence and optimality of adaptive methods on complex domains.

Development of a sharper quasi-orthogonality result.

Construction of a reliable and efficient error estimator.

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.