A posteriori error estimates for finite element exterior calculus: The de Rham complex

TL;DR

This paper develops residual-based a posteriori error estimates for finite element exterior calculus methods applied to Hodge Laplace problems, unifying the treatment of various de Rham complex cases using differential forms.

Contribution

It introduces a unified approach for a posteriori error estimation in FEEC for Hodge Laplace problems, including harmonic forms and their impact on numerical solutions.

Findings

Provides residual-type a posteriori error bounds for FEEC methods

Unifies analysis across different Hodge Laplace problems

Addresses the approximation of harmonic forms and their effects

Abstract

Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk and Winther \cite{ArFaWi2010} includes a well-developed theory of finite element methods for Hodge Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of residual type for Arnold-Falk-Winther mixed finite element methods for Hodge-de Rham Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell's equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by unified treatment of the various Hodge Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge Laplacian.